I don't know about your area, but here there are a few educational "networks" mainly based on either region or county that allow schools in their membership to post on their web servers. So maybe your local school district allows this. If not is there a PTA/PTO that might have one or foot the bill.
Hosting it yourself, if they are not going to host it in the same location as the station and not allow you to ftp to it, or will they allow ftp inside the firewall but not outside it?
Apache is a great server and easy to get going, but the key would be the bandwidth needed to view it. Again I think if your ITS dept can host and configure it on their network and allow your weather station to ftp to that site is the best inside option, and has zero cost for you.