So I just bought a Fender standard fretless jazz bass, circa 2003, from the brother of a friend. From what he has said, and from what I see, it's been sitting in their basement for a good while, so I need some specific cleaning advice.

The most prominent thing is the bridge. It's covered in grime and small specks of green stuff, and I'm wondering if I should just go to town on it with a tooth brush and some soap water, or if I should take it off and soak it in something.

Second, there were some stickers on the back of the body that I have removed, leaving a sticky residue that won't come up with my usual polish and cleaner.

Last, more of a technical question, how does one check intonation on a fretless, it seems that it would be slightly different than what I am used to?
The small specks of green stuff, might well be light oxidation from the chrome plating on the bridge. In this case, I'd advise taking the bridge unit off - being careful to note where the earth wire attaches to - so you can get right into all the nooks and crannies with a toothbrush and light machine oil.
I use Singer sewing machines' own brand light oil, and it's absolutely brilliant for this kind of job. It's cheap, shifts all kinds of crap easily, and doesn't leave a sticky residue like WD40/GT85 and suchlike will. Sick a drop or two in your machine heads while you're there.
Cheap toothbrushes are also brilliant for this type of work; the bristles are usually stiffer than good ones, and they aren't shaped.
As for the sticker residue, try boiled linseed oil. Get it from hardware shops, it's cheap and easy to use. Just remember to soak the rags you use, once you're done. Oh, and clean the oil off the neck of the bottle, or the cap'll stick like sh*t to a blanket...
Sadly, I'm not so sure about fretless intonation. I'd have to guess, find the middle harmonic by ear, then measure where the 12th fret ought to be. Put a little chalk mark, or a bit of Post-it note at that line, and proceed as usual.
The other thing that will safely take off stickers is WD40. Apply to the sticker and residue then use a soft cloth and some mild cleaning detergent to gently rub off the sticker and residue it leaves behind.

Btw, when you all decide to procreate, this method works well on wood furniture and novelty stickers as well.