Is there any specific role of imidazole in the lysis buffer used for recovering recombinant his-tag protein from E. coli?

it is advisable to add ~5mM imidazole to ur lysis buffer, to take care of any non-specific proteins binding to Ni-NTA agarose. helps in getting rid of the junk proteins at an early stage.

Imidazole is side chain of histidine amino acid and this binds to the Ni in immobilized metal affinity chromatography so when you add imidazole to the column it competes with the his tagged to the recombinant protein and protein is eluted out.
thats the basic function of imidazole in buffer. apart from it also reduced the nonspecific binding of the protein to the column.

So it functions during further purification of His-tagged protein and it has nothing to do with actual cell lysis. Right? If yes, can I add it in the same final concentration directly to the lysate after lysis and before Ni-NTA purification? because i dont have it available right now!

yeah, u can do that.....but in very low conc....~5mM...........of course, you need to add imid. during Ni-NTA purification also (at higher conc.)