He didn’t have anything to build fire with, didn’t have anything to cover his body, so he was just walking as hardfast as he could to keep warm
The last thing he remembered was that birthday party in his house, a sudden pain in the heart his chest, a choked groan and…darkness.
When he came into senses
All his body seemed to bewas covered with goose bumps.
get rid of this terrible feeling of numbness
So many times he cried for help, but with no luck
At one of the stops, he thoroughly examined his body. The reason why he did that was because he suddenly felt like there was something wrong with it. First of all, the skin did not look like skin of an old man, without all these cracks and wrinkles. It was really smooth and young-looking. Then, Peter noticed that he had went quite a distance without getting tired, which was simply impossible for an old man like him. Finally, when he had to jump over a creek on his way, he did it with suspicious ease.
At one of the stops - were there several? Why did he stop? And really the cause and the consequence are begging to be swapped:
Peter noticed that in spite of the fact he'd covered quite a large distance by then there was no fatigue which seemed impossible for an old man like him. Once he had to jump over a creek happenned in his way and he did is with suspicious ease. Something was very wrong. Finally he stopped and examined his body. His skin wasn't a wrinked skin of an old man but was very smooth and young, his muscles had the now-forgotten strength in them, his old scars were gone, and blahbhah...

resumed his ramblingwandering
where he could slakequiet his immediate thirst
ThatIt was an old little road
all covered with grass
... with grass grown through the numerous cracks.

Peter couldn’t see either tracks, or footprints on it
Peter saw neither tracks nor footprints.

Strange, but he didn’t spot any cars or other vehicles and the whole hamlet looked very poor and ancient, just like it would have been looking in the first half of the 20th century.
You're jumping to conclusions. Any other century will look the same or otherwise you should explain why the first half of the 20th century.

When it became darker
Speaking of which. What time of day it was in the beginning? I got the full impression from 'cold dark forest' that is was night. Now it was day. I can imagine nights in early summer can be cold, but certainly you won't be freezing to death in the daytime, even if you are naked.

saw the old man gazing at him with an astonished gaze.
Don't you just love being a proofreader? LOL
The old mad gave him an astonished gaze.

he himself wouldn’t have been looking any betterdifferent
'Better' sounds as if there was something bad in gazing with astonished gaze

All these candles, a kerosene lamp, rough wooden furniture, even clothes, everything shed the spirit of antiquity.
I don't quite understand what you were trying to say with 'shed the spirit'.