I don't imagine any well-built Arduino would be troubled by the sort of accelerations likely to be encountered by amateur rockets, assuming all connections are soldered.
but I'd have expected the second stage to be triggered by the first stage separating, and I'd expect that to be triggered somehow by the expiry of the first stage.
Doing it based on a timer somehow doesn't feel like a good way to synchronise the handover to the next stage.
I can't see the improvement over a very small and simple piece of timed fuse.
You can not push a fire button, but once you light a fuse, you can't un-light it if an unsafe situation develops (someone wanders into the LZ, the dog knocks over the launch pad, etc).
Most 2nd stage ignition systems are inside the rocket, between the first and second stages. It's not particularly practical to assemble stages with a burning fuse inside. The timing would be tough, too.
You need what is known as a logic level n-channel FET, connect the source to ground, the gate through a 100 ohm resistor to the arduino output pin and the drain to your load. Connect the other end of the load to the positive of your power supply and connect the negative of the supply to ground. As simple as that.
also, how bad is a rocket being top heavy?