Can attack and exhaust the turn it arrives.

Haste allows things that came under your control this turn use exhaust abilities, which means abilities that have exhaust as part of the cost (that is, the exhaust symbol BEFORE an arrow.) Haste also allows units and heroes to attack that came under your control this turn. — Sirlin
Things do NOT need haste to become exhausted as the result of an effect. For example, you can play a unit then immediately exhaust it with Boot Camp because the exhaust is an effect there, not a cost. — Sirlin