In lieu of Box, here is my reply:
You are asking the answer to two unrelated problems. Here the answer the problem about maintaining the time and date:
The battery (actually a button battery) only stores enough energy to keep the date and time. Over time, that little button battery fails to hold a charge.
It is not awfully important. I think switching it would be one hell of a hassle and a danger (You have to open up your Axim). So the easier solution would be to use
MikelSpikel's program. Use it before switching batteries.
Now here is the answer to why your Axim will not start up:
The Axim has a system that will force the Axim into sleep mode if the battery latch is opened. This is hard-wired into the Axim and cannot be disabled via software. What Box was saying was that if you fix this problem, your Axim will start up. There are various fix to this problem. So the best way is to use the search function on this forum for the answer.