I've also had the same issues, after upgrading to IOS 6.0, the apps in my iPhone 4 were crashing and the battery life run out very fast.
The only solution worked for me was to remove all mail accounts (Exchange/Gmail) from my iPhone (in settings) and then add them back.
The crashing were stopped! 🙂 also the battery was back to normal.
I found the solution by first looking at the crash logs (in IOS 6). you can find them at Settings -> General -> About (at the bottom of the About menu you will find the option that opens the logs).
There i saw many logs about Low Memory crash. 99% of the logs were pointed that the largest process (use the most memory) during the crashes is "dataaccessd".
I search in the web and found that this process is a deamon process that reponsible the Mail accounts (like Exchange/Gmail) and contancts list.
I've installed the app "System Status - activity monitor" which shows important system info like CPU usage, Memory usage and running processes.
Before apply my solution (deleting the mail accounts) i saw that the CPU usage was very high which consume the battery very fast. Also i saw that the memory (RAM) was run out.. which probably what cause the crashing..
After deleting the mail accounts (at Settings -> Mail, Contacts, Calendars) i check again with the System Status app, and saw that the memory and CPU were back to normal.
Adding the mail accounts back, still everything work ok 🙂
Hope this solution will help you too,
Enjoy! 🙂