How to solve blackberry curve 8310 trouble with "download app world"?

This most likely seems to be a firmware issue as Blackberry has moved on to higher firmware versions with the most recent devices running OS 7. Again the availability of an upgrade would also depend on how you procured the phone. If you have purchase the phone on contract via a carrier, you might need to check with your carrier on the latest firmware upgrade available for your device and update the OS to the next available version.

Of note you must be already aware that an upgrade to OS 7 is not possible for your device because of hardware limitations. However, you would be able to easily upgrade to the next best upgrade option available for your device provided the same is made available by the carrier. If you are not tied in to a carrier and have purchased a stand alone device then hooking up the device to the Blackberry Desktop Manager should help you upgrade the firmware quite easily. Believe me you don't necessarily need to be a cellphone geek to upgrade the firmware and the Desktop Manager helps you complete the upgrade in a jiffy. Upgrading the OS should pretty solve your App World issue.

Hope this helps.