Most games are compatible with 64-bit operating systems. Generally, my experience with getting older games to run on such an OS is tricky if the game is very old. Usually, the DRM has a habit of blocking the games from starting since they usually require drivers. Some older games also like to make deep-level calls to parts of the OS, or memory, that are rendered unusable in such an environment. From what I can tell while reading the net, Away Team is an exception. Searching for 64-bit specifically, in relation to Away Team, yielded few results, but those I did find are having problems like yours. I went and checked Reflexive's website
and, given the game's age, was only able to find a single post of any kind on Away Team. You could try posting there and seeing what happens, but Reflexive has pulled back from making boxed retail games, now focusing on the casual market. Activision's site was also empty of Away Team content. The oldest game I could find there, that I recognized, was Vampire: The Masquerade – Bloodlines, which is an amazing game that does work with 64-bit. Away Team was designed to run in Windows XP or earlier, so you may have to run a dual boot system.