We have managed to reach Mt. Everest (8850m) without supplemental oxygen.
However, this was done by superbly trained and ultimately fit experienced climbers.
And it was done with a support staff of thousands. Remember its not only the final climber, its not only his 10-12 person climbing support team, or the other two equally large support teams. It is also the 200 or so that stock the base camp from the nearest village.
Which village only exists because it is a supply line for Everest.
Which village is totally populated by a culture that is not only pre-acclimatized but also descendants from generations of high-altitude dwellers.
Which village is kept stocked by the proceeds of an industrial modern economy with all the vehicles and infrastructure that goes along with that.
As the OP asks, for a single, untrained, late medieval age person who is not pre-acclimatized and has no time for acclimatization..
ab.so.lu.te.ly. no chance for 9000m
9000m requires both a skilled and acclimatized climbing team, and the infrastructure to make a supply chain for the climbers.( I know its been done 'solo' and without oxygen. Did that solo climber really lug all his gear, food etc. from sealevel, by himself?)
For the 5000m in the original question... Yes. Not at all easily, but possible.
It depends on the climate and technical difficulty of the mountain.
Does your hero have access to a support staff that can build a supply chain for him?
He is a hero, we can assume he is in superb physical fitness.
What sort of protective clothing does he have access to?
Is there a local tribe that is pre-acclimatized to living at altitude, and is willing to assist our hero?
Refining the question down to just: Can a late medieval age person avoid or moderate altitude sickness, without taking the time for acclimatization?
Not really: Late medieval is 1250 to 1500 AD.
No knowledge of Oxygen, no means to make it without very heavy materials, no ways to store it.
No knowledge whatsoever of actual medical treatment for altitude sickness.(which requires tailored steroids)
Drugs that fortify the body/metabolism actually worsen your response to altitude sickness! So no opium or coke or junglefrog licking for you.