When I turn 13 years old, will some of my Roblox account restrictions unlock automatically?
No, they won't unlock automatically as such; but additional options will unlock that will become available to you, and you will have the opportunity to review and turn these features on/off yourself. This continues until your account birth date exceeds the restricted date.
Here's a snippet from the official parental support guidance:
All Roblox games feature basic security and privacy measures. All user-uploaded images are reviewed by human moderators for inappropriate content before being posted, and there are filters that weed out bad words and other problematic language. Players age 13 and older can see and say more words and phrases than younger players, but inappropriate chat and sharing personal information (including phone numbers and addresses) are restricted regardless of age. Links to YouTube channels and social media usernames can be shared by players 13 and older.
For Example: The profanity filter "On/Off" option will become available in your settings, but remain at "On" until you toggle it off yourself. Until you reach the right age, the option to Toggle it On/Off won't be available.