@dandelion
As an A4000D owner here are my tips for what to do next:
1. Remove that battery NOW! Do not wait for signs of trouble, as the first sign may be the last. Find a suitable NiMH replacement. This will require solder work.
2. I'm hoping you have a monitor to go with this machine... Otherwise you won't be able to use any of the native screen resolutions without the aid of a scan/sync doubler. Unlike the A1200, the A4000 does not have composite output!
3. Take a fine brush and a can of air to all the boards and components... Chances are you'll have a buttload of dust inside the machine.
4. Kickstart 3.1 ROMs, if they're not already installed, will likely be needed... Workbench 3.1 or 3.9 as well. Anything less is just uncivilized! :-D
5. Be very cautious about the SIMM sockets. Check for cracks and other signs of aging. The plastic used for the sockets may break off very easilly, causing contact problems. If they do break, a little super glue on the outer edges will hold them in place. Of course you'll want to make sure you have the full 16M of RAM in place before gluing them in permanently.
6. A low-profile fan on the CPU will keep it cool in that cramped slimline case. Mine is glued on with thermal tape, but I believe (correct me if I'm wrong guys) a 486 fan fits the 68040.
7. If no other joystick is available, a Sega Genesis gamepad works without any sort of modification... Tho only the B button is available.
8. Remove the expansion slot covers for extra ventilation.
Don't worry too much about Buster, unless you intend to expand this system... And even then, only if you experience problems. I have a v9 Buster, and have yet to have any problems with it (tho the only expansion card I have is the GVP HC+8 SCSI board).
Hope this helps!