Welcome to the new AP tank champion :)
I say "solo mid", but it's really just to show to you the possibility that rumble have more than a way to play him. He can be a really good mid champion, but if you are a good LoL player, you know that carry like Ashe are often more important than mage like rumble.

I have tried 3 type of build so far :
- AP/tank (1)
- Complete AP (2)
- Tanky (3)

I am gonna try the best to explain why you would want one build over the others and what these build can give you.

For each chapter, i am gonna use (1),(2),(3) to identify my 3 different build.

Legend :
- MP = Magic Penetration
- MR = Magic Resist
- AP = Ability power
- AD = Attack damage
- CC = Crowd Control (stun, slow, silence, etc.)

Build (1) : i use MP runes, defense runes and MR runes. MP runes give me more damage power and the other runes let me resist for the poor defence i am gettings in my items.

Build (2) : Same runes as the (1) but i changed defense for health/level. Why? There is no particular reason. For these 2 builds, you can switch to whatever you like between these runes.

Build (3) : MP, Dodge, MR and health. MP for magic damage, dodge for defence (can be changed for defence/level or health), MR for magic resist and health for health. All what a tank need :)

The masteries are pretty simple for me. It's kind of like a mordekaiser build. Get 9-21-0 to have AP in your offence tree and to get the AP boost in the defence tree. So because you want the defence tree AP boost, why not get all the way and have more defence.

This is applicable for all the build because you want to survive whatever the build you are going to do.

One of the biggest part.

Early game

every build start with the doran shield because it's pretty good to be able to survive. You don't really need AP at the start of the game because you will not be able to kill this early (except for gank)

Build (1) : I think that for the start you need to concentrate on 2 items, Boots of Swiftness and Rylai's Crystal Scepter. Gettings these 2 items will give you a big advantage in slowing your target and following them to kill them. If you have extra money, go get Hextech Revolver to be able to steal life with your spell. It's not a great boost but it help you survive in the lane much easier.

Build (2) : Same things as the first build but change your boot to Sorcerer's Shoes to let you kill them faster.

Build (3) : Aegis of the Legion is the first item of any good tank/support. Geth this as fast as you can. After this get Mercury's Treads begin to build your Rylai's Crystal Scepter.

Mid game

Build (1) : There is 4 items left to build and not enough time to build them all. So get what's better depending on the situation.
- Will of the Ancients if you have others AP in your team
- Rabadon's Deathcap if you know you can kill them easily
- Force of Nature if there mages are stronger than you or feeded.
- Void Staff if they are getting too many MR
you should normally be able to get 2 of these items. choose wisely.

Build (2) : same thing as the first build. The only difference here is Zhonya's Hourglass that will give you more armor. So get it against AD team. Lich Bane is a great item but if you spam your spell too much, you will silence yourself when you don't want to. So be aware of your heat when you have this item.

Build (3) : Same thing here.
- Sunfire Aegis : it's a great "i don't know what to get" item for tank.
- Force of Nature if there mages are stronger than you or feeded.
- Zhonya's Hourglass : give you more defence against AD.

Late game

Finish your build and go win the game...

Here a list of other items that are good for different situation.

Against CC : Banshee's Veil
Against AP : Abyssal Mask
If you die too much : Guardian Angel
Alternative of Will of the Ancients : Hextech Gunblade
For tanky build or juste to slow them : Randuin's Omen
Agaist AD : Thornmail
For tanky build : Warmog's Armor
For more melee damage : Lich Bane

* Lich Bane is good but it overheat you very fast. So be carefull when you fight with this.

In all my builds, i go for Electro-Harpoon first and boost it the fastest i can. after this i boost my Flamespitter and my Scrap Shield.

When you engae combat, try to slow him first. after that go on him and use your flamespitter to do massive damage. If you want to do more damege, dont forget to begin your attack while being in the danger zone.

Keep your shield to save your *** or to chase the ennemy. it's not worth to use it when you engage.

Your utlimate can be use to engage or to chase. due to its slowing effect, it's nice to shoot it first so that you won't miss your target. And if you don't use it to engage, it's great to slow them while they are escaping. His long range give him a great advantage.

2 spells, Exhaust and Flash

There is many other spell but these are the best.
Exhaust is just great in every situation. In addition with your slowing power, no one can escape or get to you (if you are 1v1)

Flash is the best spell to engage/escape any situation.

Pros

Build (1)
- Balanced build that can survive and kill

Build (2)
- AP power that can rivalize with many champion

Build (3)
- The most durable build. It let you CC your ennemy to let your teamate kill them all.

Cons

Build (1)
- He is still squishy if you engage too much.

Build (2)
- Fully squishy
- squishy melee is often the first target of team fight

Build (3)
- Not a lot of Damage power.
- Many other champion are better suit to do the tank job.

Due to my skill sequence, i am less good at farming in the begining. But it let you play against any ranged champion without being in danger. You will still be able to farm good enough by using your harpon and finish them with your flame when they are close to your tower.

The part you actually want to read.

Yes you can easily solo mid with Rumble.

Just spam your harpon on the creep and on the ennemy champion. Harass them with all your might. You don't have any mana and your cooldowns is pretty fast for this skill (like 7 second in the first level)
Practise your skill shot and own the mid. When you are good enough to kill them, shoot your ultimate and finish them.

Dont be afraid to use your flash to close the range and shoot all your skill on him.
Even if you go in overheat, you do bonus damage so kill him fast.

Its also a double edged-sword. Don't go crazy and think you can take turret shot like the god. Take your time and set your pace.

So finally, you are The tanky mage... go take anyone slow them, get them angry, so much that they will want to chase you and then BOOM! you gank them.

Give your comment, good or bad.

Thanks to have read this build.