Welcome from one old guy to another.
I don't know about this for certain, but I would think having a subwoofer behind a solid wall would profoundly limit it's performance. All sound, but particularly low frequency sound, is merely waves of moving air. If the sub can't move the air in your room, you aren't going to get much bass response.
If you can tell us what brand and model of subwoofer you have, we'll be able to tell if it's sealed or ported which will have some bearing on our responses.
If your sub is a sealed sub (no port) and the driver faces forward (is not on the bottom of the sub facing toward the floor), or if it's a ported sub with the port on the same cabinet face as the driver (like the Axiom EP500), placing it behind a hole in the wall covered with speaker cloth might work OK.
If it's a ported sub with the port on a cabinet face other than the one with the driver, yet again, you would suffer some degradation of the sub's performance because, even though the driver would be moving air in your room, that port would be moving air in the other room.
Any possibility of placing the subwoofer near field (close to your seating area)?
From an
article by Alan Lloft, Axioms "resident expert."
"
Use the subwoofer as a nicely finished end table or support for a lamp or plant.
EP500 Cherry Natural - VaSSallo Series
Subwoofers—the big boxes required for ultra-deep bass—don't have to be ugly. You can order one in a highly attractive real-wood finish and locate it at one end of a couch as an end table to support a lamp or plant. The versatility of surround sound acoustics allows for the subwoofer to be placed almost anywhere in a room because we don't hear “direction” from low bass energy. So the subwoofer does not have to be at the front of the room near your front speakers, screen and video display. As often as not, it may in fact perform better at the side or the back of the room or, in some rooms, off in a corner."