It depends. If the servers are identical (same distro and same version of that distro) you can copy files. If the servers aren't identical, it depends (copying between distros is likely to fail, between versions of the same distro may or may not work).
The easiest way to do it is to copy the entire repository (after running
configure && make), and then just run
make install on the target system, just like you suggested.
Usually mono installs into /usr/local by default (it depends a bit on the system), and if you don't have anything else there, you should also be able to just copy that directory to the target system (if you do have something there, you can create a file list before and after running
make install do have a list of files you need to copy).