Can I use Airport Express to repeat wifi without physical access to the router?
Your Airport Express will only work as a wireless extender/booster if your main access point is an Apple Router, like another Express or an Extreme or Time Capsule. 

However the way around this for a non-Apple router is if you hard-wire it (run an Ethernet cable straight from your modem or router to the Airport Express). 

This might be not applicable to your situation since you do not have physical access to the router.

There are other (more universal) Extenders on the market, for example from Netgear (for about $50). I use one, and it works like charm in extending existing wifi network that I do not have physical access to.