So I got a Quest to capture a Double Premium horse. What is the cheapest/fastest/etc way to accomplish this?

Try buying a Map and Cross your fingers

Maps tend not to work for me. When I get those quests I wait until the next capture event (some events do refunds like capturing ISH on St. Patricks Day) and capture 30-50 horses. I usually get a double premium horse. Given that even one Adventure capture map goes for about 20k it's not even more expensive as you'd still need to pay for the capture as well.

Thanks for the suggestions! I completed a different quest and got a Capture 400, so I'll have lots of oppurtunity to catch a good one haha.

Do the Maps guarantee at least P? I know it adds 10 or 15 points, but what does that calculate out to be?

In my opinion they rather make the Sub-pars a Superb but they don't guarantee a Premium.