ok well it is possible, however somewhat risky if you dont know what your doing.
You can put a 3rd party firmware on the router called dd-wrt. WARNING:!!! this will void your warranty, failure to follow the instructions posted on the site for your model may result in bricking or completely destroying your router. If all directions are followed correctly, you should be fine. But continue with caution. Also you can not go back. once you flash it, you cant go back to the normal firmware I have to say that because i did brick one of my routers out of the 10 or so i did so there is a possibility. If you need help, ask before moveing on. better to spend 3 days preparing then 1 dead router
However if you get the firmware on there you are able to set the wireless to a wireless bridge which will do as you say and make the router connect to the network and you get to then use the lan cables as lan cables. Also the firmware is very good. has many great features that linksys does not add
I also have to add that you must make sure it is ok to borrow your neighbors internet.