First, arsenal spec is not a good 1v1 spec. Tracer missile is your best abilitiy and you have to be immobile to use it. Not saying you can win, but most skilled Operatives will have ya on the ropes if nto dead.
As a Pyro, I see allot of operatives and this is normally what I do:
1) Operative stealth attacks you (IE: knock down/4 second dps burst). You have to eat it and thank goodness for heavy armor and high expertise (higher the better).
2) Hit Jet Boost. (This will give ya some space to open with your rotation.)
3) Move away as far as you can, but within ranged of your abilities. Hit Incendary Missle/Thermal Detonator/Rail Gun. (Massive burst here). if you follow up with Unload, then you are allowing the Operative to close on ya. Only use Unload if they are low on HPs.
4) Continue to "kite" the Operative since they have ZERO closing ability. Use Rapid Shots so that Combustible Gas Cylinder procs.
5) If Operative is either trying to heal or below 50%, then Stun them. The pop your relic/stims and repeat #3 for the kill.
I never have an issue if my abiliites are up or health above 75%. Operatives are masters of ganking folks that are usually under 50%. So, don't let your HPs go below 50% otherwise you are asking for trouble and probably death from a skill Operative.