How do airports manage gate assignments for aircraft?
Depends on the airport. Some airports assign blocks of gates to specific airlines, and they then assign them to flights (usually on an as-needed or first-come-first-serve basis).
Other airports assign them all themselves, again usually on an as-needed or first-come-first-serve basis.
Of course there are categories of gates as well, size and capacity determine what aircraft types can be handled by specific gates.
In theory at least, and no doubt small airports do it that way, it can be handled all with pen and paper. Larger airports of course will have some sort of computerized system for it, either purchased off the shelf or something they had custom built for them.