Probability that the President would win election against a random adult citizen?

Yes. The politician wins. Focus groups and polls are clear on this, they won't vote for random people. Trump would also beat them easily.

(Obviously some probability you get someone effectively not random from a random draw, but it's very low.)

Random person most likely has an IQ of 100 which is a standard deviation or two below a random president. Random person most likely has less talent and experience at politics, promotion and attack than a politician. Random person is most likely less physically attractive, less charismatic, and less wealthy than a politician. Random person doesn't have a gigantic support apparatus behind them (although even if they do they're probably still screwed because it's not enough to make up for the rest of the deficiencies). As others say it won't even be close. Random person will predictably be slaughtered. 

Not a very principled answer, but: 98%

There is already a substantial preference for incumbents (about 65-35 I think), and I think this would be much stronger if the challenger was completely unaffiliated with politics (I want to say something like 90-10 if challenger and sitting president were equally electable otherwise, maybe 75-25 if the challenger is actually substantially better than the president, 99-1 if they're just average).

Say there's 5% chance they're equally or more electable, 2% they're substantially more electable. Then there's 0.5% on them being equally electable and winning, 0.5% on them being substantially more electable and winning and 1% on them being less electable but winning anyway.

So 98% overall.

Edit: didn't consider that the major party might organise in support of the candidate, this changes things a lot. Say there's a 20% chance that the out-of-power party campaigns for the rando (40% the rando is somewhat-aligned, 50% conditional that the party can get their shit together to campaign for them); then the incumbency advantage will be more like the standard 65-35, assuming equal presidentiality (but I still think they're probably a lot less presidential)

All told...this gets me to 96% chance the sitting president wins.

Politicians are good at campaining. Most people are not. A moderately successful politician at the national level would shred through any random people.

To add to the consensus— if a random person actually had a favorable matchup against a career politician of any stripe, that would be a massive low-hanging fruit for existing political actors to capitalize on. The RNC and DNC would be falling all over themselves to present "average joe" candidates if doing so provided a consistent advantage. They're not, so it follows that either those organizations are both highly un-optimized for winning elections (probably not; too much money at stake) or else that the evidence doesn't bear out that they should.

The VAST majority of adult citizens would decline rather than suffering the scrutiny of the public, and the unpleasant work of internalizing and repeating the lies needed to win, especially with the expectation that it would get even worse if they actually won.

General Sherman had this one right.  

There's enough magic/counterfactual assumptions in the question that I don't think I can provide a probability - it's a different universe than I live in, so presumably has different voters and different populace.  

Current president of Poland went from being relatively unknown parliament member (I'd bet, less than 5% of general public knew who he was? Also there's another public figure with the same last name, so maybe even some people who knew the name were confusing him with the other guy) to being the president within one year. So with longer timescale, big support from one of leading political powers, and some political backlog it is possible. There are surely similar stories around the world, of outsiders rising to leaders. Of course what you're asking is much more extreme, but that provides some staring datapoint.