Unless you look really young, I doubt you will have a problem. Just walk tall and be confident, most of the time they are just pretending to check your boarding pass and card, they aren't doing a detailed look at it. Be persistent, if they say no, tell them you just want to get a coffee and check your email, not drink.

As others have suggested, post your travel dates here, someone is sure to be able to help, they don't even need to be on the same flight. ^

Good for you if you've discovered the joys of airport lounges and business class travel at a young age, enjoy.