How to reset my password for my @me.com mail account?

You can change your password at http://appleid.apple.com as long as you know the current one to sign in with.

If you've forgotten the password, go to http://iforgot.apple.com and sign in with your iCloud login. A new password will be sent to your associated email address. If this doesn't work you will have to contact Support. Go to https://getsupport.apple.com; click' See all products and services', then 'More Products and Services, then 'Apple ID', then 'Other Apple ID Topics' then 'Lost or forgotten Apple ID password'.