No matter what the fingerboard may be made of, I would use the mildest method possible to start, and only progress as far as necessary into stronger methods.

So I'd start with a dry cloth, then damp (but not wet) cloth, then maybe window cleaner, and then on to alcohol or naphthalene (lighter fluid).

My reasoning would be that firstly the mildest methods are preferable since they have less risk to the fingerboard or the rest of the instrument.  Secondly would be that even real ebony is sometimes color treated or stained to enhance it's appearance.  Many pieces of ebony are not evenly dark, and while that won't necessarily harm their functioning well as a fingerboard, it is possible that a lighter streak in the wood was stained a bit to give a nice uniform appearance, so solvents and aromatic hydrocarbons like naphthalene might take off some colour.  Using solvents like alcohol or lighter fluid also could dry moisture out of the wood.  I couldn't say for sure if it would be enough to increase the chances of cracking, but I would be at least as careful about putting them on wood as I would be of getting the wood too wet.

I figure you probably threw away the applique?  If you haven't thrown it out yet, then you could experiment on the removed applique to find out what will remove the adhesive on it. 

I can think of one thing I would try before maybe anything but dry cloth.  Tape.  Adhesive usually sticks to adhesive better than it sticks to anything else.  So I'd try cutting a little strip of something like duck tape and stick it to the areas where you have adhesive on the neck in very small sections and pull it off quickly and see if it lifts off any of the sticky residue.  Kind of like using tape to remove lint from a garment.  It shouldn't damage the wood to try.  It may not work either, though.  I haven't ever tried it for something like this.

Well, Kevin, you and others here are more experienced than I, and so I note with much interest the preference for alcohol as a cleaner. I hadn't considered that most of the stains may not be alcohol/spirit based. 

My own experience is more with guitar fingerboards and assorted electronics.  Other than the dust and dead skin, there the usual "gunk" next to the frets may be oil if they ever actually wipe down the fingerboard with anything.. But often is actually sugar/starch from food.  While it could come from particles in the air from cooking and etc in a home, most often it is from playing without having washed or wiped off their hands after eating.  So I usually try a damp cloth with mild soap first before going to alcohol.  Sometimes it is enough. People don't typically put tape on guitar fretboards, though, so the adhesive traces aren't anything I've seen there.  With my electric violin, as the tapes came off, I just rubbed the bit of adhesive off.  A minor pain in the butt, since it wasn't a lot of adhesive from those thin strips.

I have seen ebony fingerboards even from reputable companies where after some wear or cleaning, lighter colored bits of the grain became visible.  While it is possible that never is done on violin fingerboards, I would kinda doubt it.  But I will definitely admit I have little idea what actually might be used on violin fingerboards as stain base, and you and the others here definitely have the experience on that.