I don't like the white rabbit as a little old man, so I made him a young hipster. (Because hipsters are the whitest, youngest people I know
Unlike Denise and Alice's sheets, this one says less about his personality. So I'll note it down here:
The Rabbit hates to be kept waiting but is always late (to work, to parties). Instead of carrying a watch, he carries a smartphone. He wears fingerless gloves because there aren't cool enough looking fingered gloves that let him use his phone's screen.
He is a sycophant to the Queen of Hearts and a servant to the Duchess. He ignores Alice or treats her as a nuisance until it looks like the royals have an interest in her. Only THEN does he treat her with courteousness.
Alice was attracted to him because she loves rabbits, but once she catches up to him, she realizes he's not that great.