At our shop, we suggest a new cassette every 2 times you replace your chain, IF you stay on top of your maintenance schedule. With 99% of customers this would mean a new cassette every 1-2 years depending on how much you ride it. If you are a commuter and ride everyday, your timeframe for replacement gets much smaller.
We would also assess the condition of the bike. If it looks like it has been neglected, we can, with high confidence (we do hundreds of these bikes a year), tell that you would need a new cassette as well.
What I tell every customer in your situation is as follows:
A chain that is not cleaned regularly will attract and hold dirt. That dirt will act like 50-grit sandpaper on everything it touches. When the chain stretches out, it will wear the cassette equally. If we try to put a new chain on a worn out cassette, the teeth on the old cassette will match the old worn out chain and won’t properly hold the new chain. We recommend replacing the cassette for every 2 chains so we can guarantee shifting performance from our tune. If replacing the cassette is something you want to hold off on we can try to put a new chain on the old cassette, however we may find that we can tell what your favorite gears are and it may skip under load.
You can always ask them to just put a chain on, check to see if they can make the adjustment, and call you if it is skipping or poorly shifting.
However, since your chain and cassette were only a month old when the chain snapped, unless you have a bent/broken tooth on your cassette, it should be good if everything else is adjusted correctly.
As far as the snapped chain, it may come from manufacturer defect, improper installation, or you may need to revise your shifting strategy.
You want to make sure you’re not cross-chaining. This will put a lot of stress on the components and will wear things out quickly and possibly snap your chain. Also, your bike doesn’t have a clutch, so you have to back off on power when your shifting. Right before you’re about to shift, let off pedaling a bit and wait for the gear to change. This will increase shifting speed and accuracy.
We do see a lot of customers resolve shifting problems with proper information on shifting. (If you’re shifting the proper way, then disregard. It’s more for people who will view this post later.)