Accessing administrative tools in someone else's network is going to require assistance/approval from the network administrator (which you indicated isn't you).
If you suspect that someone has gained unauthorized access to your network, then the best thing to do is change all the access codes. Before proceeding with this step, disconnect internet access and shutdown wireless, then after changing all the access codes, passwords, scanning for (and removing) any SpyWare or viruses, etc., then it may be okay to get connected again. Contact the network administrator for help with this, as they'll probably want to know that there's some unauthorized access occurring.
For a shared network, my first guess regarding unauthorized access is that a previous user may be using an access code that hasn't been changed for a long time. If codes are changed regularly, then either someone's deliberately not keeping this information secret (e.g., they're sharing with friends who may be sharing with others, and so on), or someone has SpyWare that's publishing the access codes somewhere.
As far as finding out the routing table, you could use "traceroute" to see where traffic goes, and then there are "ARP discovery" tools that might help identify other connections too, but the best way is likely to check the router's reporting options that show you which hardware addresses are using the network (this might be limited to DHCP-assigned addresses only though), but you'll need assistance from the administrator for this.