What you should have is FTP access to the site which would allow you upload files using a program like FlashFXP or WSFTPro or similar. Also that way you can put a password on the account and only give out the login details to people you trust. Oops forgot to add you can upload more than one file at a time.
If however you can't do any of the above, you'll just have to upload one file at a time.
BTW Here's a free host for you 100MB limit, unlimited bandwith..........so loads of people can download at the same time, fast speeds too!