The well is not in every level, and I didn't notice a pattern to it appearing. To be able to drink from the well, you must first purchase the Cup of Life in a shop, which costs 399 gold. It does not appear in every shop, so if you have the gold, buy it.
Once you purchase the cup, it stays with you for the entire run. If you die, you lose it. After you buy it, take it to a well the next time you see it, and walk up to it. A bar will refill your health, and once completed the trophy will unlock.