iPhone 4 Internet Problems
Try enabling your airplane mode for 10 seconds. Then disable and try accessing your App Store/apps. If this does not fix it. Do a hard reset. 

Hold close iPhone button/sleep and the home button, till the screen blacks. Wait a few seconds and then hold the close iPhone/sleep button to turn it back on. (iPhone 4's top button)

If you don't trust the hard reset idea, google it. 

Hope this helps. :)