A growing number of hotels and guesthouses have wi-fi internet connections, but usually only in public areas, rarely rooms. Remember also that this is not an island of high-speed connections. For (slow) public wi-fi, lurk outside the Rodrigues Regional Library in Port Mathurin.
Most of the island's banks are in Port Mathurin, though Mont Lubin and La Ferme also have banks, and there's an ATM at the airport. There's also an ATM at the police station near Plaine Corail.