If tanisma : acquantance, then what is tanismayi and tanisirim mean ?

Thanks in advance.

tanışma is acquaintance. "Tanışmayı" is accusative case of "tanışma".

Tanışırım is conjugated case of "tanışmak" verb for 1st-singular pronoun.

thanks.

So in this sentence  ?:  o zaman seneye tanışırım artık would be something like : when I met him years ago ?

No. Let me explain you word by word:

o zaman: then, so 

seneye: next year

tanışırım: I meet (forgive me, I haven´t expressed the tense of conjugation. It is for simple present tense)

artık: from this moment. (but, in this sentence, "artık" is used to emphasize the meaning of any time at future)

The meaning of sentence is;

So, I meet him next year.

Literally (with combination of simple present tense + future tense emphasizin word "artık"):

So, I will (or I may) meet him next year.

I hope I could explain. Ask if there is any point that you did not get.

thanks.