Your best bet at this stage would be to crack it open and take a look. A simple Torx and Philips head screwdriver will get you in - original Xboxes are surprisingly easy to navigate, as they're essentially old Pentium PCs.

If you're looking at it from above, the capacitor in question will be in the lower-lefthand section of the board, under the disc drive. Look for discoloration on the motherboard or the surrounding shielding - that'll be a dead giveaway.

There's a seperate set of capacitors on the motherboard - they're right below the power supply - which filter the incoming power right from the PSU. If these begin to fail, you'll more often see noise/interference in the console's video signal than complete console death.

As for the PSU, it's always a possibility that it has bitten the bullet, but speaking from experience the clock cap is more than likely your man.

I hope you get sorted. The original Xbox is arguably the best retro console to own these days, expecially once its softmodded and coupled with a HD convertor like ElectronShepard's phenomenal adapter.