Migrating applications without using Migration Assistant?
It really depends on the application. With the advent of the Mac App Store a lot of applications have become more self-contained (everything lives within the app bundle). But, you're talking about an older systems so again, it just depends on the application.

The easiest thing to try is just copying them over from the old /Applications folder. If that doesn't work then your best bet is to re-run the installer (assuming it's still compatible with 10.8). If your old Mac has a disc drive you might be able to use the Remote Disc feature to complete your installation.

I know some app developers have started to make digital copies available for people who find themselves without a DVD drive anymore- but you might have to reach out to them for help.