I would hope it's fairly obvious you won't be able to host it 24/7 without the machine running the server switched on... If you want this, a dedicated host is definitely the only way to go. Forget who I used to use but it was something like $5 a month. It's maybe not a tiny amount where you are but for most that's pretty trivial.
Edit: I used a company called "Mulitplay" for dedicated servers. Thinking on, I don't recommend them. I upgraded from like a 5 seat server to a 10, they started charging me for both (the 5 and the 10). I pointed it out, they
ed me around. I went to my bank to do chargebacks, they got pretty arsey, I got arsey back they came back with a sob story about one of the big tornado's in the US causing them problems... I'd avoid them
As far as the dynamic IP goes, there are a few services that can be run from a PC. They update a server with your new IP and it is able to redirect traffic to your router. It uses a protocol called "dyndns" (have a google around for that an you'll get a lot of info)
If it's at all feasible (money... sounds to maybe not so easy for you) there's a whole bunch of routers that have this ability (to plug into a dyndns server) as part of the management page. You MAY even be able to get a 3rd party (i.e not from the manufacturer) firmware that opens this possibility. I've never had to look for it but do some googling for "router tomato firmware" and check if your router can use it, there's probably something in that to counter highly dynamic IP addresses.
Best of luck