Congrats on the new button!
A pic would help determining his age - but if he is more than 2-3 months old or so, it's not really possible to tell his age, in my experience. If his chin has clearly defined black and white markings (not present in all color mutations), he is likely to be older than 2-3 months.
A thing you should know about button quail, is that they usually don't do well when kept alone - they can actually die, from stress I assume, if they are alone for too long. So if I were you, I would start looking for a hen for him right away.
With regards to bringing him in for the night, you are probably doing the right thing. My buttons are exposed to freezing temps, but they have been outside since summer, so I know they are acclimatized. And there are people on here who says buttons shouldn't be exposed to freezing temps, acclimatized or not - I live in Denmark, our buttons might have been bred a little more hardy than buttons in the US, for instance. But this is my first winter with buttons, I can't say for sure they'll be able to handle it yet - we are only just beginning to get extended periods of frost.
You might want to consider placing this 'larger cage' inside, if possible, to avoid having to move him all the time - in general, buttons don't appreciate handling. If the 'larger cage' is not suitable for standing inside, you can build one that is, buy a rabbit cage for him or use a large plastic storage box or similar.
What exactly is this food you bought for him? In general, game bird feed with 20-30% protein is recommended, but they can eat seed mixes and similar too, though you might have to supply him with vitamins and an additional protein source if he only gets a seed mix.
Do ask, if you have more questions, I'll be happy to try to answer ^^