Well is your friend physically directly above you? You DON'T want to run the cable outside since it's rather dangerous electrically and I'm sure your building doesn't allow that sort of thing. If he's physically directly above you and the apartments are identical then it will be pretty easy. Just get yourself a power drill and a drill bit large enough to shove the end of an ethernet cable through, drill in a convenient place, and there you go.
If you decide to drill just make sure to avoid areas close to electrical sockets, phone jacks, etc.
If you want to share your internet connection over wired ethernet you'll need:
any standard cable/DSL 2+ port router (most come with 4 ports)
NICs in both of your computers
possibly a small switch at your friend's place
Drill your hole and run a cable to your friend's place. Plug that cable into a small switch OR get one that's long enough to run straight to his computer. I'd recommend getting a cheap 5 port switch for the convenience of it. You'll have a long cable going through the floor that you can leave alone in a closet somewhere where it won't get tripped on or pulled ever, and he'll have the option of putting his computer wherever he wants and connecting up any length cable to the switch in his closet or wherever you drill the hole. That'll also leave him the option of adding more computers up there later without the need of running another cable (or buying the switch later). Plug your net connection into the WAN port on the router, plug your computer and his computer's NIC cables into the LAN ports on the router. After that, just configure the router though it will likely work for both of you after you plug it in with no need to reconfigure.
I'm not recommending you drill simply because it's probably against some lease agreement and you don't necessarily know what is in the ceiling between you and your friend. If there's a decent amount of space between the floors drilling will be a good bit harder. Wireless is a safer bet if you're inexperienced with this stuff. Wired ethernet will be cheaper, and it will be faster though. If you think it would be "safe" you might want to ask the building management about the possibility of having their maintenance people do the drilling for you but I doubt that they'd do it.
Gaidin