How to fix mundo so he can be a healthy champion but still able to go where he pleases:

Just do two things:

1. Remove the 20% current health cost on mundo's ultimate
2. Make mundo's ultimate health gain rather than health restore

Here's the detailed breakdown:

Mundo activates ult and begins gaining maximum health over time: up to 40 / 50 / 60% of his total health over a period of 12 seconds.

Because it is an increase to his maximum health and not health restore, spirit visage wouldn't improve his ultimate-- but by the same token, grievous wounds wouldn't diminish it.

If he had 50% of his health remaining and his ultimate is increasing his maximum health by 5% per second, his current health would effectively only increase by 2.5% per second because his current health will only increase proportionally to his %remaining health.

Because the ultimate no longer has a 20% current health cost and only increases his current health in proportion to his remaining health, Mundo ideally wants to activate his ultimate at full health, and doesn't have to play in such a way that causes him to be in danger of being executed before he has a chance to benefit from his ultimate.

As an increase to his maximum health, his E and any other sources of max health scaling will be increased (especially near the end of his ult where his max health really begins to ramp up)

Being a maximum health increase, blade of the ruined king and other %health scaling damage becomes a very attractive option for countering mundo, but isn't as much of a hard counter as grievous wounds is for him currently.

On the flip side, mundo ultimate doesn't allow him to sustain in lane anymore and once it times out his health returns to normal. If he activated ult at 30% remaining health to use the movement speed to get out of a jam, he'd be at 30% remaining health when it ended.

Being a maximum health increase, even if mundo takes a lot of damage his allies can potentially overheal him beyond his normal capacity because his "health cap" increases over time during his ultimate.

The more his allies can keep him near full health, the more effective his ultimate will be at increasing his current health-- it becomes progressively more difficult to do that as he really begins packing on the max health cap towards the end of his ultimate, so the first 4 or 5 seconds of his ultimate is critical.

Mundo now has a way to fail to become a tanky god if he misplays the early seconds of his ultimate but he can still have some influence from range with cleavers and doesn't have to straight up die. Compare that to his current ult, where if he misplays the first few seconds of his ult he just gets executed.