Retrofit apartment for gigabit internet
First step is to get an 802.11ac router. You're probably going to want wireless anyway, so just do that now. Live with that until you decide either you are happy with it, or you need to upgrade. And remember: you won't get ac speeds unless both the router and your device are ac.

If you do need upgrade to wired, buy 2-4 small gigabit switches. Terminate all of the wall plates with cat5e rj45 jacks. At the wall plates with 2 jacks,  hook up both jacks to a switch. Since you have daisy-chained drops, the switch will bridge the two halves of the chain. Those cost of doing this should be under $100 for switches and wall plate parts. This is a less than ideal network configuration, but since your bottleneck is going to be your internet connection speed, you will likely hit the limits of performance on this network.