Well, now it gets complicated.
The RT-N13U supports both router and repeater mode, but they're mutually exclusive. Once you switch to repeater mode, you lose all the very features that make it a router; NAT, firewall, your own private network, etc. Repeater mode is quite literally just that; it repeats the signal in a form that's identical to the original. And if that signal is unencrypted, only allows one device, shared by others, etc., all the same things apply to the repeated signal.
What you really need is a WISP (wireless ISP) router. In that case, you’d have a full blown router (NAT, firewall, your own network, etc.), but it would connect to the ISP over wireless rather than a wired connection from the WAN. So in that sense you’d be repeating and routing on the same device.
But alas, you don’t have that option. For you it’s one or the other, not both. But if you had an ethernet port in the room, then of course you could use it as a traditional router. The other option would be to add a wireless ethernet bridge and connect it to the WAN, effectively creating your own WISP router. Or perhaps install dd-wrt (third-party firmware) and configure it as a WISP router.
As I said, it gets complicated.