Tough luck man...that's a non-standard case you've got there. What's the specs on that? Pentium 1-2 I bet. Nice monitor though...gateway EV series, right? Those are good.
I'm having a hard time finding a pic of the guts of the 4500, but I believe at that point in time they were using mostly mATX mobos. That may or may not be a BTX case. It could even be that funky clamshell design like they used on the optiplex line. (though the larger ones were still mATX)
The other problem with Dells (and several others) is that they use a proprietary connector for the switches and lights. (power button, HDD light, etc) However, with a bit of work you can make that connector work. I've done it.
It's not really a big deal to run caseless though...just make sure the mobo is sitting on something that isn't conductive. Anything paper, cardboard, or wood will work admirably.
edit: Going by the pics that show the 4500 next to monitors, I'm guessing it's going to be mATX at most.
edit again: This video doesn't show the case, but it shows the mobo from a 4500. mATX it is.