In my totally non-medical opinion
I reckon if you're getting ANYTHING out at all (3 drops, 1 drop...), your breasts aren't broken. Hand expressing colostrum really is the way to go as there's so little of it to start with. Also you're getting a bit of skin to skin contact there too, even if it's just your own skin!
If you ask for small syringes, make sure you've got the right ones! I ended up getting some from a MW friend that I filled with colostrum antenatally and ended up not being able to use because they had some sort of coating that wasn't going to be safe (or something like that!)
When I was doing that, my BFC friend advised to do 10 minutes on each side, by doing 4 minutes on each, 3 minutes each, 2 minutes etc so you don't get too uncomfortable, but hopefully things will get going. If I were you, I'd be trying to do that probably every 3 hours at a minimum, but obviously you have other things that you'll be wanting to do as well (like having cuddles, if you can), so just take it as easy as you can.
It's horrible having a LO in NICU. Is he on CPAP at the moment?