Yeah I've done Horrible Hilly Hundreds full distance twice.
(https://www.horriblyhilly.com/ride-details/)
Man that is tough, and only about 35% of the way there. Now, a lot of the distance/time comes from transits between the various climbs, but man. I guess if I spammed a single hill in that I'd guess Everesting would be the equivalent timewise/effortwise of about that plus 25-50%.
So you pick a hill to do it on. First of all, get a triple and the biggest rear cog you can. THe key to non-pro-bike-rider climbing is getting lots of gears. For hills all road groups are hilariously too coarse for real sustained climbing. Lance Armstrong, pinnacle of the EPO era, was famous for high-rpm cycling and let's face it, his sustained Wattage output (to say nothing of watt: weight ratio) was 2.5 to 3x what an everyday cyclist could sustain. And yet he probably ran the same 53/39 and 11-25 groupset that they sell at the LBS. Yeah he was spinning a bigger gear, but the point is HE'S 3-4x THE POWER/WEIGHT RATIO and on the same gears.
So get a triple (52-40-30) and go to MTB on the rear if you have to to get to 30+ rings on the back.
Then, weight weeny everything else. Especially your wheels, climbing is about imparting and losing angular momentum to the wheels as much as the actual overcoming of gravity. Protour riders used to have certain ultralight rims IN THEIR CONTRACTS insisting the team provide them regardless of their equipment sponsorships.
Now, what's the best hill to pick? This is all about long term endurance which means Zone2/Zone3. It's 8800m. so...
5% grade: 20x vertical height so that 180 km/112 mi.
10% grade: 10x vertical height so that's 56 miles.
7% average is probably the sweet spot. Not so slow that you're losing your mind, Not so fast where air resistance comes into play.
You'll want a hill with a far steeper descent that ends up in the same place.
You'll want a hill with smooth concrete. Once you start getting tired, every crack, bump, and hole becomes misery and momentum killers. Bike climbing is about sustaining momentum.
But finally, you'll want a hill with a tailwind. Oh yes, it matters a lot, every percentage point is very viscerally felt when climbing inclines. You'll generally be sitting up in the saddle, and a tailwind that is >5mph will be PUSHING YOU, with your body as a sail.
If I could find a 8% Grade climb with a 10+mph tailwind on good concrete a well setup bike, this is probably easier than doing an Ironman (2.4 mile swim / 112 mile bike / 26.2 mile run) on courses like Ironman Wisconsin (where the bike is basically Horrible Hilly lite).
I'm assuming you get a refill/refuel provision/support? Don't carry excess liquid and food!
Finally, it will help to lose weight obviously.