How long does it take to earn 2,000 gold on Clash Royale?

As you can see here, the gold contens of a chest very much depends on the arena you currently participate in. You say it is Arena 3 (Barbarian Bowl), so that's what I am assuming throughout the reset of the answer. Higher arena means more gold, so faster gold-getting!

A silver chest will give you (averaged) 38 gold, and a gold chest 127 gold. I disregard the giant and magic chest, see that as a bonus if it pops up :). There are 180 silver and 54 gold chests throughout the whole cycle (source), so roughly 3 silver chests per 1 golden chest. If you open them in order (that is, SSSG), you'll be netted (38*3 + 127) = 241 gold in 17 hours (3*3 + 8).

However, there are also free chests and the crown chests, that add to this amount. You'll get roughly 4 free chests in this time period, and 2/3 of a crown chest. I'm not sure how much this is, feel free to comment this if you are on arena 3 and open a free/crown chest, I'll add it in. This depends on your playstyle: if you play only a few games per day, not enough to win the crown chests, it takes longer. If you play just once a day, you can only open 2 free chests, instead of the theoretical maximum of 6.

Furthermore, you can donate cards in your clan and get money (and experience) in return. How much you donate is completely up to you. As told here, it is a maximum of 4200 gold per week. You'll get 5 gold for a common donations and a staggering 50 gold for a rare donation.

In conclusion, it differs quite much per player. If I can get the gold per crown/free chest, I will calculate the approximated time when you play at maximum efficiency.

Technically Yes. It's a good card and I bought it myself as I wouldn't get a drop. But I've stopped using it recently in favour of the balloon as it would get less distracted, and accompanied with freeze does more damage. Moreover they will be nerfing it on the 24th reducing charge speed with 13%. If you don't have many damage dealers, the prince is a good choice, in a good, big, active clan its easy to get 2x2k a week by donating so you can buy plenty of cards on the market.