This feature is available with some paid plans
The TTL (time to live) in your DNS settings can be left at your domain provider's default settings. Your custom domain's TTL should not affect your Bitly links.
GoDaddy has a great summary of TTL:
"TTL is the amount of time servers cache the information for your DNS records. For example, if you set the TTL for a particular record to one hour, servers store the information for that record locally for an hour before retrieving updated information from your nameservers."