I want to bleed my brakes and am looking for any advice on using a syringe to push brake fluid into the caliper and up through the line through the master cylinder.

My question is this, will the fluid push through the master cylinder and up into the resevoir or does the cylinder prevent fluid from backflowing that way?

 

It would work, but its a pricey tool to do that.
Part of the reason the brakes need bleeding, is to get the moisture and particles of rust or other junk out. And because of gravity, all the particles end up in the caliper, so the shortest distance, for the particles to go, is out the bleeder.
I just gravity bleed them, to do a complete exchange of fluid, and then have one of the kids help with the bleeding using the brake lever to push the air thats left over, out the bleeders.