On the train Lupin looks tired, ill and sleeps through almost everything that happens meaning he was completley drained of energy. That makes me think there must have been a full moon around the start of term. He was probably too tired to arrive at the school in the typical manner for teachers, which we are never told exactly what that is. Also, I'm sure the professors arrive a few days before the start of term, but if I were Lupin I would have wanted to stay home for the transformation.

It's also possible that Dumbledore encourage the idea of him taking the train both so he could rest, and keep a close eye on Harry. Dumbledore might not have been anticipating Dementors, but Black was supposed to be a madman. They all figured he killed 13 muggles and a wizard in broad daylight, what's to stop him from breaking onto a trian full of school age kids? Lupin probably lept at the chance to protect his best friend's son.