Yes. You can drink it before the month is out but it may be cloudy as the yeast sediment has not settled yet.
Ideally one week for primary fermentation in the bucket, one week in the barrel in the warm for the priming sugar (which you will add to the barrel) to ferment and carbonate the beer. Then 3 put the barrel in a cold place like a shed or garage to condition for 3 weeks. Conditioning lets the yeast sediment drop to the bottom and solidify and also results in better carbonation.
Be careful not to shake the barrel about to much as you want the yeast to settle.