You don't need to have access to the router. You need access to a location where the wireless signal strength is sufficient. There, you can position a repeater or something similar to boost the signal.
The best option would be to use an actual repeater. Like this:
http://www.linksys.com/servlet/Satellit...name=Linksys%2FCommon%2FVisitorWrapper
The problem is that they only work with a few wireless routers.
A slightly more flexible option is to use a service called WDS. This service connects multiple wireless access ports together so it seems that they operate on one network. Not all wireless routers support this either though, so you will have to check out the existing wireless router in the apartment.
This last option is guarenteed to work, but requires two wireless router connected to each other. One of the routers would be configured in client mode and connect itself to the apartment's wireless network. This router becomes a wireless bridge, joining the existing wireless network to its own wired network. The second router works as usual and creates your own wireless network. The second router is then connected to the one that is in client mode. Basically, the client mode router acts as an internet connection that the second router uses to distribute internet access to your own wireless network.
Of these three options, WDS and repeating are the simpliest, but require support from the existing network. The bridging technique can be difficult to setup because most wireless routers don't support it directly and require that you upgrade it with third-party firmware that does. The Linksys WRT54G is one that does (after firmware upgrade). You could avoid the difficulties of setting up a wireless bridge if you bought an actual preconfigured wireless bridge (Linksys makes some, but I don't know their price).
In terms of price, WDS would probably be the cheapest (routers that support WDS are fairly common). Repeaters are usual more expensive than regular wireless routers (maybe double in price). The bridging setup of course requires two wireless devices.
EDIT: Don't forget that for any of these solution to work, you must have access to a location in which the signal is already reasonably strong.