Have you got a lathe, or access to one, or know someone who can machine a few basic parts up for you?
If so, piston tanks are fairly straightforward. If not, they're a lot more difficult. Which isn't to say they're impossible to make with handtools, but you will need to be very competent at bench fitting.
Hobby servos tend to rotate at between about 50-100RPM. For a syringe sized tank you would only be using a fairly modest threaded shaft with a smaller pitch, so with a servo you'd be looking at a minute or so to retract the tank, which is ridiculously slow. I would aim for a filling speed of between 10-20 seconds. you can go quicker, but this generally requires larger motors, which can be bulky.
You can get small geared motors on ebay which will run very happily on servo electronics, and if you're happy to tinker with things, you can use the electronic board to power the motor, giving you a very inexpensive and capable proportional dive tank. Generally standard servo electronics will drive a motor up to about 2-3 watts. Some high power servo electronics will drive considerably more than this.
A couple of minus points with syringes, firstly the pistons are often rather large, this means they take up more space than a custom machined item would. Second point is that the seals themselves tend to have quite a bit of compression, more than is actually optimum for our use, which increases the friction somewhat. Neither of these are major issue, but it's definitely worth taking into account.