The TTL (time to live) in your DNS settings can be left at your domain provider's default settings. The TTL should not affect your Bitly usage.
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.
Keep these things in mind as you choose and set up a custom domain:
- The domain for your short links must be different than your website domain.
- It can't be more than 32 characters long, including the dot.
- If you haven't added one yet, Bitly offers a complimentary domain. However, if you add your own domain, you won't be able to redeem this offer later.