The last human alive is on the Antarctic continent and needs to find a way to leave the planet before Earth becomes unsustainable for life. (He would look for a planet orbiting a younger star. That target planet may have been discovered and selected by astronomers some time ago but nobody has actually been there. Let us suppose the travel time would take five hundred years or so)
For the sake of this question, suppose:
- That man can't die, so it's fine if it takes him ten thousand years to build the spaceship. I mean that he is fundamentally immortal and whenever there's a risk for some fatal accident or even desire to kill himself, something will happen to prevent it. He does not age and does not fall sick (or if he does, he will eventually get better). In other words, if in our universe some event would have three possible outcomes A, B, C with probabilities $p_A=99.9\%$, $p_B=0.0999\%$ and $p_C=0.0001\%$, and A results in certain death for the character, then in that world, actual probabilities become $p_A'=0$, $p_B'=99.9\%$ and $p_C'=0.1\%$.
- Runaway greenhouse effect has made the continent pleasantly warm in summer. They are covered in trees and populated by small rat-like mammals, but parts of the planet away from the poles are arid and unbearably hot.
- He could travel around the planet to find raw materials if necessary, but he'd need a way to protect himself of the heat.
- His body has been modified (artificial skeleton, etc). He has the strength and weight of today's average sixty year old man, not more. He can carry light objects, he needs to eat and drink mostly like a regular human being.
- The old research centres (and, elsewhere on the planet, ancient cities) are still there but have been abandoned for a thousand years, so there's no electrical device left in a usable form.
- He has an engineering and physics background. Based on the immortality principle, if happened to choose the academic education that would be helpful for him for this endeavour.
- The story takes place in the far future so we may assume that physics unknown today have been discovered, and new technologies have been mastered and are well understood by that guy. (No fancy space-time modifications or teleportation, or time travel. At most more efficient propulsion methods, or energy harvesting techniques.) However any actual artefact has been lost.
At this point of this story, he is growing plants, cooks his meals on wood fire and collects rainwater.
My idea is to replicate the technological development, skipping parts that are not necessary, and my main concern is how to bootstrap this development. To obtain raw materials he needs to dig into the ground (wikipedia mentions significant amounts of iron, as well as natural gas and oil), so he needs an energy source. Should he go for wind or water turbines? Is it possible to find in the ground materials for generating and maybe storing electrical energy? Once you have machines that can dig into the ground to collect any raw material, supposing he knows how electronic devices and fusion reactors work, does it become feasible to construct them given enough time and patience?