Yes, it applies to all movements

Unarmored Movement (PHB, 78) States:

Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield.

It does NOT qualify as your Walking Speed (or Swimming Speed, or Flying Speed)...just Speed. This applies to all types of movements and would add additional speed to the Fly spell as well.

This is supported by Jeremy Crawford as well.

Unarmored Movement is intended to increase a monk's innate speed, including an aarakocra monk's flying speed.

He further clarifies Bonuses and Penalties that apply to Speed in general:

Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc

As the Unarmored Movement does not specify a specific type of speed, it applies to all.

Fly Spell specifically

Fly states (PHB, 243)

The target gains a flying speed of 60 feet for the duration.

The implication here is that you (the target) had no Fly Speed (or a lower Fly Speed) before casting and it is now a speed that you, the target have. Once it becomes your speed, it becomes subject to Unarmored Movement.

Similar to Special Types of Movement in the PHB, page 182)

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.

Here, Swimming has a penalty, unless they have a Swimming Speed. Once they have that Speed, from whatever source, then Unarmored Movement should apply.

Rule of Cool

And it's fun! Monks are supposed to be super quick. Let them be super quick in all their speed!

Here are the rules on falling from the Basic Rules (or PHB p. 183):

A fall from a great height is one of the most common hazards facing an adventurer. At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.

The fall itself wouldn't cost movement (it's "forced movement", except in this case the thing forcibly moving you is gravity). However, if you take damage and thus land prone, it will cost movement equal to half your speed to get up afterwards:

Combatants often find themselves lying on the ground, either because they are knocked down or because they throw themselves down. In the game, they are prone. You can drop prone without using any of your speed. Standing up takes more effort; doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to stand up. You can't stand up if you don't have enough movement left or if your speed is 0.

To move while prone, you must crawl or use magic such as teleportation. Every foot of movement while crawling costs 1 extra foot. Crawling 1 foot in difficult terrain, therefore, costs 3 feet of movement.

Assuming you have a speed is 30 feet, if you spend 5 feet of movement to step off a ledge, you will take 1d6 bludgeoning damage and land prone. Then it will cost another 15 feet of movement (half your speed) to stand up and continue moving; alternately, you could remain prone and continue moving, but you'd only be able to move 12.5 feet. (Your DM may have you round down to 10 feet if you're playing on a grid map.)

However, if you are for some reason able to land without falling prone or taking damage, then you could move 5 feet and fall off the ledge, land on the ground below, then continue moving as normal.