Yes, and yes. A better barrel than stock may not necessarily take longer to heat up, but they seem to maintain their accuracy better. A heavier barrel will simply take longer to heat up, but at the cost of weight.
ETA: I'm assuming your rifle is still in its original Remington plastic stock as well? A better made barrel, from what I understand (and will be corrected if I'm not) will heat pretty evenly or uniformly, as opposed to a stock barrel, which might get hotter in certain spots of the barrel. A better stock that allows the barrel to float freely will help, but a better barrel SHOULD maintain its accuracy, even when its relatively hot. You might experience a POI shift, but should maintain its accuracy, though you don't really want to keep throwing rounds down the tube as quickly as possible if you want the barrel to last. Some people are in the "shoot a round a minute and you're good" camp, others are in the "shoot a five round group and let it coll a bit" camp (assuming you're not a competitive shooter, where time is of the essence, but you'd probably be using a better barrel to begin with).