The detailed question is:

Is there a word $w$ on which a TM M $M$ halts after a maximum of $|w|$ (word length) steps?

I highly assume, that this problem is not decidable because in the worst case you have to test every word that exists (infinite) to realize, that the TM never holds. However this isn't a proof and I have to proofe this question (without the Rice's theorem).

My idea was to use the subroutine technique to "convert" the problem into the halt-problem, the $\epsilon$-halt-problem or the total halt problem. I have no idea on how to turn this problem into an existing one - could you help me here please?