You can't do anything with Leopard on iCloud except access email, but to do that you would have to set the iCloud account up in a compliant Mac or device, you can't do it in Leopard.
So, yes, you need to update to Lion or Mountain Lion, provided that your Mac is capable of handling them. To make matters more complicated, you can't access these except from the Mac App Store, so you will first need to upgrade to Snow Leopard - you can get that in the Apple Store (use the Search).
The requirements for Lion are
- Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or Xeon processor
- 2GB of memory
- OS X v10.6.6 or later (v10.6.8 recommended)
- 7GB of available space
I'm not sure whether you can still get Lion - it's not in the Mac App Store so you would need to ring the Apple Store and ask (the number is at the bottom left-hand corner of this page).
The requirements for Mountain Lion are listed here:
If you are able to thread your way through this fandango you can reactivate your @me.com address by signing in in System Preferences>iCloud, though you will not be able to recover any emails previously held on it, nor indeed any data which was in your MobileMe account. You will get an iCloud with that address plus the same thing @icloud.com, and you can use the account to sync calendars, contacts etc. between devices.