Hi - if you just have the one red key then all that will do is lock and unlock the doors manually and start the engine. Any branch of Timpsons will be able to clone it and I really recommend that you do so - just in case you lose it.
The alarm system is entirely separate - so no - it will not arm when you manually lock. I would guess that, if an alarm system is fitted it will be the Meta system you will see an LED on the fuel cap/headlight adjust panel.Normally you get a round black fob with a button in the middle - press it and it locks the doors and sets the alarm. These fobs can be bought from here:-
Meta Remote controls
I bought a spare one from here myself. HOWEVER you will need either the 8 or 16 digit code that came with the original and for that you will need a working spare fob which, I guess, you do not have. The key you have will not help here.
If you got some documentation when you bought the car you may find a red, rectangular piece of plastic called a Code Card. If you have then the code will be the one needed. HOWEVER - without a working fob you still have a problem. If you have this number I would suggest you call Abacus and ask their advice.
Personally, in your situation, apart from getting a spare key to use in an emergency, I would not bother. Our GTV is an import and has no alarm or remote central locking fitted. I do not feel it is insecure and the insurance company knows and has not charged extra. We have had it for 8 years now and I have never felt the need to alarm it and the lack of remote locking is not an issue.