Depending on your router face.
If it is circular or has a flat edge, you can follow a straight edge. There are lots of different edge guides. Metal ones are generally better. Even a outside edge from a sheet of plywood could work. Often the guide edge cannot be clamped in the middle of the work. Screwing a block over the guide can be useful in preventing the edge from lifting.
Not having tried it, two edge guides might be better still eg bracketing the router so it has to stay on the line.
If not, you need a guide attached to the router that can follow a straight edge.
The biggest issue is you need to start with a plunge cut is the "rabbet" does not run the entire length. Of course, depending on your carpentry skills and need for perfect, you could make the cut all the way across the piece.
I would start the plunge away from the end and move toward one end. Then go the other end. A good vacuum will make everything easier. Also masking tape on the outer edges should (might) reduce splintering.
The other question is how deep a plunge. This will depend on the type and size of the bit you are using. You might want to experiment. For instance a 3/4" plunge and rabbet most likely is too much for your bit and router. You might try 1/4" or 1/8" at a time.
Also for 1 5/8", might have to do multiple parallel cuts. I am not sure if you should do each to depth, then move to the next one OR do all the same depth. The router should have a good footing on the surrounding (uncut) surfaces either way.
Having said all this, you have posted an image. You are looking at the edge of the wall which may or may not have curves.
If it is truly a straight cut, I would use a table saw. You will still need to chisel out the ends.
If following curve, you would use the router to cut the inner most edge first, as there is the most support for the router face. To cut further, you need a really large face extension (
https://www.rockler.com/universal-router-base) OR this one (
https://www.rockler.com/mpower-combinat ... -base-crb7) OR you must put wood against the outside edge to support the