first off, nice car. I love old bimmers; I got a '86 325 myself that i plan to drive until the wheels fall off, glue them back on, and drive it some more.
![Grin Grin](/yabb2.2/Templates/Forum/new/grin.gif)
now, the battery light should mean that its not charging right or the battery is shot. i think auto zone or advanced auto can test both of those and charge the battery. thats a good place to start.
I also know that the e30s (like mine) have an issue where a single-shielded wire to the reverse lights runs past the shift linkage and can get worn through after time, causing a short. this drains the battery and just sucks in general.
so if its not the alternator, then it could either be a battery that won't hold a charge or a short. test for shorts by pulling the fuses and checking resistance across the circuits with an ohmmeter with the ignition off and the battery disconnected. some circuits should have a small leak current, like the clock and radio to maintain presets. but if his won't last a day, its probably a big leak or a crap battery.
best of luck!