Understanding the Thickness Difference
The first thing to understand is that original New Orleans plaster walls are not the same thickness as modern drywall. Most plaster-over-lath assemblies are about 3/4 to 7/8 inch thick from the finished surface to the back of the lath. Standard drywall is 1/2 inch thick. That leaves you with a 1/4 to 3/8 inch difference that you have to deal with somehow, or your patch will sit recessed below the plaster surface.
You have three options for handling this. The first is to shim out from the framing with thin strips of plywood or wood lath so the drywall sits flush with the plaster surface. The second is to use 5/8 inch drywall instead of 1/2 inch, which reduces but does not eliminate the difference. The third is to install the drywall recessed and build up the surface with joint compound to bring it level with the surrounding plaster.
I have used all three and they each have tradeoffs. Shimming is the most work but produces the best result. Using 5/8 inch board is intermediate. Building up with compound is the easiest but only works for relatively small differences and tends to crack over time at the transition.
Cutting Out the Damaged Plaster
Use a utility knife or oscillating multi-tool to cut the perimeter of your patch area. Cut all the way through the plaster to the lath behind it. You want a square or rectangular hole with clean straight edges. The cleaner the cut, the easier the patch.
Try to position your cut so that the edges of the hole fall over wood lath, not in the gaps between lath strips. Lath gives the new drywall something to attach to. If your cut falls between lath strips, you will need to install additional backing later.
Once the perimeter is cut, remove the damaged plaster from inside the cut area. The plaster will come off in chunks. The lath behind it may or may not stay attached. If the lath is in good condition, leave it. If it is broken or rotten (common in water damaged areas), cut it out and expose the studs behind.
Wear a respirator for this work. Old plaster dust contains all kinds of things you do not want in your lungs, and in homes built before about 1980 there is some chance of asbestos or lead in the original plaster or in old paint layers. If the plaster is from 1940 or earlier or you find loose flaking paint layers, consider getting a sample tested before cutting it aggressively.
Setting Up the Backing
If the existing lath is intact and runs across your patch area, you can attach drywall directly to it with drywall screws. Use 1 1/4 inch screws so they penetrate the lath and slightly into the stud behind. Drive them carefully because lath is brittle and will split if you force the screw.
If the lath is missing or damaged, you need to install backing strips. Cut 1x3 or 1x4 wood strips long enough to span across the back of the hole, with each end resting on solid lath or framing on either side. Screw the strips in place through the existing plaster around the hole, using long screws (2 1/2 inches or longer). The strips give your new drywall something to fasten to in the middle of the patch.
If you are shimming out to match plaster thickness, attach your shim material to the studs first, then everything else gets shimmed off that. Use construction adhesive and screws for shims. The shim layer is the foundation of getting the patch flush with the plaster surface.
Cutting and Installing the Drywall Patch
Measure the hole carefully. New Orleans plaster walls are often not square, so do not assume the hole is rectangular even if you cut it that way. Measure all four sides.
Cut the drywall patch slightly smaller than the hole, about 1/8 inch smaller on each side. The gap gives you working room for the patch to sit without binding, and you will fill the gap with joint compound later. A tight friction fit usually means the patch sits proud or angled because you cannot quite get it positioned right.
Screw the patch in place with drywall screws every 6 to 8 inches around the perimeter and into any intermediate backing. Drive the screw heads slightly below the surface of the drywall, leaving a small dimple but not breaking the paper face. The dimples will get filled with compound later.
Check that the surface is flush with the surrounding plaster. Run a straightedge across the patch from plaster to plaster. If the patch sits recessed, you will need to build up with compound. If it sits proud, your shimming is too thick and you need to back off. Adjust now, before you start mudding.
Bridging the Transition Joint
This is the step that determines whether the patch is invisible or obvious. The joint between the new drywall and the old plaster is a substrate transition, and it needs to be bridged in a way that handles the slight differential movement between the two materials.
First, run a thin bead of acrylic latex caulk along the joint line, working it into any gap with your finger. Caulk gives a flexible base under the tape and helps prevent cracking at the seam over time.
Next, apply paper tape across the joint, embedded in setting-type joint compound (the 90 minute kind works for most patches). Paper tape is better than mesh tape here because the bond to the plaster substrate is mechanical from the paper fibers, not just adhesive like mesh. Mesh tape on plaster has a higher failure rate in my experience.
The first compound coat over the tape should be wide, maybe six inches on each side of the seam. Feather the edges out so the compound transitions gradually onto both the new drywall and the old plaster. After this coat dries, apply a second coat slightly wider, then a third finish coat in regular all-purpose compound.
Sand carefully. The old plaster around the patch may have texture or imperfections that you do not want to flatten. Sand the new compound to match the surrounding plaster, not to flatten the plaster to match the new compound.
Priming and Painting for an Invisible Finish
Even a perfect physical patch will look obviously different if you do not handle the priming correctly. New compound and old plaster absorb paint very differently. Painting over both without proper priming gives you a halo effect where the patch is brighter or matte compared to the surrounding wall.
Use a high-build primer designed for transitions between substrates. Ask at the paint store for what they recommend for plaster patches. PVA primer alone is not enough for this application. The high-build primer fills the slightly different surface textures and gives you a uniform base for the topcoat.
Apply primer not just over the patch but feathered out 12-18 inches beyond the patch into the surrounding plaster. This prevents the visible boundary between primed and unprimed areas under your topcoat.
For the topcoat, paint the entire wall corner to corner whenever possible. Spot painting almost always shows a halo around the patched area because slight sheen differences between old paint and new paint catch the light. Painting the full wall costs you a quart of paint and an extra hour, but it eliminates the most common giveaway that a patch was made.
When the Patch is Going to Show Anyway
If your existing wall has texture, the patch is going to be more challenging because you have to match that texture across the seam. Most New Orleans plaster walls are smooth, but some have a sand finish or a swirled finish that was applied as the final coat of plaster.
For sand finish, sprinkle silica sand into your final topcoat of paint before applying. The sand becomes part of the paint film and creates a matching textured surface. Practice on a sample board first.
For swirl or trowel marks, you can mimic them in the final coat of joint compound before the compound dries fully. A damp sponge or trowel worked through the surface can produce comparable swirl patterns. This takes practice and you should do test patches in a hidden area first.
If the existing texture is too distinctive to match (some older Orleans homes have very particular plaster finishes that the original plasterer specialized in), the honest answer is that a perfect match might not be possible. You can get close, and an experienced eye will know it was patched, but a casual visitor will not notice. That is sometimes the best available outcome.