Whats a good router for apartment

First I would use a wifi scanner to determine what wifi networks you have around you, and what channels everyone is running on. Hopefully you have an Android phone or tablet, Wifi Analyzer (link) is a great free tool to do this with. 1, 6, 11 are non overlapping channels. If you have people on channels in between both of you will have problems. If you have say a bunch of people on 1, 3, 6, you would want to choose 11 to reduce interference.

Also where do you have your router placed. Ideally it should be central in the apartment, and preferably up higher. Microwaves, cordless phones, etc can also cause problems, so preferably keep the router away from things like this.

Are all your devices 2.4 GHz only? If you do upgrade, it maybe worth considering switching to 5 GHz if possible. The 5GHz band is much less crowded, and there are 40 non overlapping channels to choose from. Very far distances aren't always as good, but you won't have that issue in an apartment, or most house for that matter.

What would really help is an wifi router with external antennas. In my experience these just always perform better. I recently moved, and had to replace my ageing WRT400N because the internal antennas just couldn't compete with all the other interference. Upgraded to a WRT1200AC, which on sale is a solid router for the normal price it becomes more of a questionable decision.

So if you decided to replace it, suggestions would depend on your budget. There are several routers in the middle of road area that offer good performance without being ridiculous expensive ($300 for a router is insane). As with any of these do some research, but I would check out:

ASUS RT-N66U - $117

TP-LINK Archer C7 AC1750 - $99 (Actually several in the Archer line that usually have very good reviews)

Just watch, as some AC routers don't always have the best 2.4 GHz performance, but most will still be much better than your internal antenna router.

As for your modem, with your plan, I wouldn't worry about upgrading it. I believe the problem is your wifi. As for it locking up or resetting itself, I've seen several of these do this. My solution for this has to always just place the modem (and usually the router) on a timer, so they both turn off in the middle of the night, and come back on a minute later. This "reboot" always seems to help prevent problems during the day when you are actually using it. These timers are cheap and can be picked up anywhere. I have used these in the past 2 Outlet Digital Timer - $15, mine came from WalMart, but their site apparently is having problems at the moment.