Sealing open space behind kitchen counter in rental with shoddy rehab?
This gap is too large to fill with caulk alone, but you can fill it and then caulk over the top. Using rubber tubing is an OK idea, and should work, but there are other products which are specifically designed for this purpose.

You can use foam backer rod, also known as caulk saver to fill the space behind the counter top. It comes in many different sizes and you should find one that will fit that gap. The benefit of the backer rod is that it can be compressed more easily to fit tight areas. It usually is also a cheaper solution.

To hold the rod in place before caulking you can put a few dabs of caulk on it to keep it from dropping down too far if certain areas of the gap are too big. You should position the backer rod about 3/8" to 1/4" below the lip of the counter so there is enough space to put a good layer of caulk down without having a raised bump there.

I would recommend a silicone caulk that is specifically designed for kitchens and bathrooms. Also, try to find one that matches the color of the counter top as closely as possible. A trick to get a clean line of caulk is to put masking tape on the wall and counter and leave an even gap. Squeeze out a bead of caulk, and then use your finger to smooth it out. Wait about 5 minutes, and then carefully remove the tape. This should leave a clean line, and there shouldn't be any excess caulk.