Assuming your contract includes data (if it doesn't, doing the following may either not work at all or cost you a small fortune), you could enable mobile data on your phone (Settings | More .. | Mobile Networks | Data enabled).
Alternatively, you could use WiFi. If your computer getting internet via WiFi, you could connect to that or try somewhere like a coffee shop that provides free WiFi.
If neither of the above are possible, you could download the apps to your computer then move them to your phone via a USB connection.
Some phones come with sync tools that can help - e.g. Kies if you have a Samsung, HTC Sync if you have an HTC. There are also sync tools available on the Play Store.