I have lost 2 relationships similar to yours when I told them the truth at the begining. At first the girls say it doesn't bother them but when their friends and family finds out, it leads to problems and girls value the opinions of their female friends more than anything. If they think you are too old for her, you are done. So if you want to keep her, go with the white lie and say you're 24. But realize that one day she will find out. So it is always good to reveal your true age to her voluntarily at the right time. Usually the best time to tell her the truth is when your relationship is mature and she tells you she loves you. When a girl is fill with love, she'll forgive you but it's important that the truth comes from you and not another source.
That's how I would handle the situation. But I have to say I don't condemn the honest approach, it is just more risky. It's your call.