Ah, I missed the "Airport is stopped". In any case, the ability to wake the system is trigger by a carefully crafted packet -- not just any packet will do. This is because systems can receive network traffic constantly. If just any packet could wake a system then many systems would effectively not be able to "sleep" at all.
In any case, you would need a fairly recent version of an Apple Airport Extreme base station (with recent firmware). Another computer wont send the correct wake-on-network packet, but the Airport will send it if it sees traffic attempting to reach one of the Bonjour-registered services.
If you don't have an Apple router that supports this, there is another option -- slightly more cumbersome...
You can get a utility (there are many of them) which allows you to send the wake packet. E.g. here's one I found over at versiontracker:
http://www.versiontracker.com/dyn/moreinfo/macosx/24421
I have not personally tested this myself. Also note that it doesn't specifically say it supports 10.6 (although for purposes of what such a utility would need to do, it probably doesn't really matter.)
With an Airport, you don't need to know if the computer is awake vs. asleep -- just use it and the Airport base station will handle it. Without the Airport to handle this automatically it's up to you to use the utility to manually wake the remote computer before attempting to connect to it with anything else.