The description of Unarmored Movement:
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield. This bonus increases when you reach certain monk levels, as shown in the Monk table.
At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during the move.
Essentially, the improved Unarmored Movement feature allows monks to treat a vertical surface as equivalent to a horizontal one for the duration of the move. So unless it's icy or something and you'd need to make a check to move horizontally across such terrain, you wouldn't need to make such a check when moving vertically across similar terrain.
However, as the feature states, the ability to avoid falling only applies during the move. You can not stop your movement to do anything while on the vertical surface unless there's something else keeping you from falling. If you do stop... you fall.
So there's no way to climb a 1000-foot wall without falling unless you have a very high walking speed.