Can I play Switcheroo from Hidden with only 1 unit on my battlefield, which is Ember Monk? Would Ember Monk's ability proc, or would it fail because there is no valid target for Switcheroo?

To answer your question: No, you cannot play Switcheroo in this scenario, because it requires two units at the same battlefield to function.

Here is the breakdown of why:

1. The Legality of Playing Switcheroo

According to the card text, Switcheroo's effect is: "Swap the Might of two units at the same battlefield this turn."

To play a card, it must have valid targets for all of its instructions at the time it is played. Because you only have one unit (Ember Monk) at your battlefield, there is no second unit for Switcheroo to target. As per Rule 358.1, you must check that all chosen targets are legal before the card is played. Since you cannot fulfill the requirement of choosing two units, you cannot legally play Switcheroo.

2. What about Ember Monk's Ability?

Ember Monk's ability reads: "When you play a card from [Hidden], give me +2 might this turn."

Because Switcheroo cannot be played (due to the lack of targets), you never actually perform the action of playing the card. Therefore, the trigger condition for Ember Monk's ability is never met.

Summary: You cannot initiate the play of Switcheroo, and consequently, Ember Monk will not receive the +2 Might boost.