I want CB to automatically login the user when he/she opens his/her browser.
You can't. CB has no idea who they are. The only way to keep track of that is using a cookie, but that would be a beyond massive security issue. What happens if someone is on a public computer? They entire account was just compromised. Your best option is to use "Remember Me", which does this to an extent (the cookie has a lifetime). Most browsers will clear this cookie anyway so it really won't work well in either use case.
I am using CB on my Joomla server for our intranet. Is the only way to solve this, perhaps the usage of LDAP integration? Already all my users do exist in CB/Joomla. Will turning on the LDAP integration integrate with the existing users, or will it add extra users? Bare in mind, that the user names in CB are the same as the user names LDAP.
It'll probably want to create new LDAP users, but I can't say for sure; I suggest giving it a try. It still won't auto log them in at any rate as they'd need to input their LDAP credentials.