This is actually quite a common problem I've seen with writers just starting out - if I had to hazard a guess as to why it happens it's because as we first learn to write language the focus is very much on constructing the sentence not the paragraph (or other larger unit). When you start getting larger units of text the flow of the whole becomes more noticeable and prominent.
So how do you go about preventing this happening? By playing a little looser with the individual sentence construction and using the context to infer the words that you want to avoid overly repeating.
Taking your example:
That left the final option. Using the tools within his cell. Looking around, he mentally slapped himself. How could I be so stupid he thought, staring at the 4 chairs set around the table. He stood up, and walked towards the table, grabbing a chair and repositioning it in his arms. He returned towards the stairs, and started climbing, walking along the rim, to avoid breaking the fragile aged wood. He paused at the second step, lunging forward to avoid the third step, the split pieces of wood, now hanging limply from their supports on the side. Reaching the top of the steps, he started swinging the chair, the legs aimed at the door.
The first use of the pronoun (his) can be replaced like so:
That left the final option. Using the tools within the cell.
You see, we (the reader) know it's "his" cell, he's the one in it! And you can do the same throughout the passage - the context of the surrounding text, combined with the point of view will allow the reader to understand who is doing what action.
For example this:
That left the final option. Using the tools within the cell. Looking around and seeing the four chairs set around the table, he gave himself a mental slap. How could I be so stupid? He stood, walked to the table and grabbed a chair before returning to the stairs and started to climb. Stepping carefully along the rim to avoid putting too much weight on the aged, fragile wood. The third step was split, the pieces of wood hanging limply from the sides. But with a brief pause on the second step before a controlled lunge over the gap, he reached the top of the stairs and started swinging the chair, legs-first, at the door.
Cuts the he/himself/his from 9 to 4, but it's still clear that "he" is the one doing all this - since there's only one person involved in this sequence of events so we don't need to reference them much since it's not changing and the reader can assume that they follow on. I'd actually hazard a guess (given it's a cell of some kind) they are the only person even present - which gives you even more leeway, no-one is going to be wondering who it was who picked up the chair or who climbed the stairs.