Can my 2005 MacBook upgrade to Lion?

It looks like the first MacBook came out in May 2006, so let’s assume that’s what you have. You can upgrade a 2006 MacBook to Lion, but that’s as far as you can go (Mountain Lion will refuse to install). However, there’s no telling whether that will fix the problem you’re having with Mail. You might want to try searching for troubleshooting steps for that independently. One option is to install a different email client, e.g. Mozilla Thunderbird (a free download).