If it's a brand new battery, it should in theory start the car and run it for a while. As the battery is hardly used when engine running. If it's not starting at all, but as you said arming the alarm, it could be a starter motor problem.
In my opinion I would do a checklist of easy things to check.
1. Have you got fuel in the tank.
2. Is the air filter clogged.
3. Have you checked the car status with an OBD connector.
See if you can get your hands on a multimeter, set to read 12v then disconnect the battery terminals and wait about 15-20 minutes for the battery to stabilise. If it's under 11 its probably on its last legs.
You can check with terminals intact, just will get a slightly lower reading due battery being use.
Do you know the age of battery?
To put how much power starting a car takes, my battery set with a voltage reader, reads 12.4v when I turn it over it drops to 0.70v so you can guess it really hampers the battery, but only for a brief second then the alternator kicks in.
Thinking about it, could the alternator be on its last legs too?.
Looking at online videos and experience, it looks easy enough to do it, just to buy a spring compressor is probably the same cost as for it to be done at garage. Depends if it needs replacing, as if I did it, I'd want to replace them both.