Sharing your exact age is up to you, but I would keep it private. Sharing that you're 22 could seem a bit cavalier early in your tenure.
Things you can do while you settle in to the role:
1. Focus on demonstrating your capabilities. If you're smart and do good work, your colleagues will notice and quickly respect your opinion and leadership.
2. Get creative in avoiding the topic. It's better to avoid the topic in a fun or funny way than to give the impression you're offended by a question about your age. If someone says "Gees, you look young!" reply with "Thank you!" If someone asks "How old are you?" reply with "Somewhere between 5 and 50 - hard to keep track."
3. Confront individuals who inappropriately comment on your age. If someone consistently brings up your youth in a way that makes you uncomfortable, confront them. Let him/her know how his/her jokes or questions make you feel and make it clear you'd like them to stop. Some individuals unknowingly cross the line with what was intended as a harmless jab.
You might also consider removing graduation years from any public profiles (e.g., LinkedIn) as these are a proxy for age and I'm certain someone will eventually look you up.
No-one but the HR department needs to know your exact age. If you're ever pressured to provide your age when you'd rather not, alert your manager or another trusted leader in the company.