It's possible to be in love with someone after knowing them for a month.
There are no rules to this stuff, and no advice anyone else can offer you that's better than what's in your own heart and head are telling you. Love, attraction, lust, friendship, amusement -- it's all so individual, it defies pigeonholing. Over the years, I've learned to lie back, relax, surrender, and enjoy the ride.
I've had a "long-distance" relationship for years now. We spend more time apart than together physically, but strangely I always feel that my partner is around. That's us. Does it work for everyone? Of course not, and guess what? I don't care.