According to Everymac's MacBook Pro listing, there is no such thing1 as a MacBook Pro 13" Late 2008 5,1. The earliest 13" MacBook Pro was a June [Mid] 2009 5,5. The smallest Late 2008 5,1 model was only 15". To clear things up:
- Maybe you mean MacBook and not MacBook Pro?
- Is it a Unibody model, or Pre-Unibody?
- Is it really a 13" or 15"?
- Is your battery integrated? That is to say, can you easily remove the battery without opening the case up using a screwdriver?
Please clarify your question, and specify the correct MacBook [Pro] model. If you could provide a link to the model on the Everymac site, that would be useful. The more accurate the information you provide, the more accurate the answer. As Kevin Grabher implies in his answer, each model is different, and behaves differently.
Regardless, I have been running a 17" 2.4GHz MacBook Pro Mid/Late 2007, Santa Rosa model, 3,1 which for all intents and purposes as the same as an early 2008 4,1 (power wise) without a battery for two years, without an issue.
1 If the Everymac site is wrong, and you do indeed have a 13"" MacBook Pro 5,1 then I apologise.