Technically you don't need a URL, but it makes things a lot easier than remembering your IP. You would need to port-forward port 22 from your router to your machine, as your public IP will technically hit your router first. You can use a service like DynDNS to get a URL that points to your IP constantly - otherwise, on most ISPs, your IP can change at any time, unless you pay for a static IP. You can get a free account and would have something like "ravenbar.no-ip.com" (if it's available), instead of having to remember your IP.
If you are planning on opening up a port for something like SSH, be sure you lock it down really well. Aside from having an extremely tough password, I would make sure you configure it to only accept SSH over SSL. Having your SSH tunnel hacked is about the worst thing that can happen to you as a Linux user, since that would allow someone a way to control your system and do anything with it. Even if you don't store anything of use on it, it can easily be turned into a vector of attack to the rest of your network, or as a botnet for other malicious intentions.