Gyumri's historic town center is quite walkable, and fun to explore, but beyond that it is a bit of a sprawling town, and to see everything it has to offer in a day, a car is needed. If you haven't come with your own, a taxi in town, with a meter, is probably your only realistic option, and a pretty good one at that. If you care to try and figure out the numbers of the local van routes marshutni, you can have quite an adventure. It may be hard to make it to some of the fringe sites though in a marshutni, like the fortress.