Hi Fragile - this needs a bit of work, I think.  It's quite hard to follow what you mean in places.

 Dear Sir or Madam,

Motivation Letter for ..... project

I would like to apply for ..... program.

As I am going to graduate from high-school next year, I have begun thinking about different universities both in Lithuania and abroad. I believe that a person cannot trip up in choosing his future. [Don't understand what you mean here] All my life I have been interested in Maths but I have never wanted to study it, that is why I have chosen to relate my life with [settled on?] Economics that looks more attractive to me. [This sentence doesn't quite make sense - do you mean that Economics is more interesting to you?] 

There are  many reasons why I would like to participate in your program. First of all, the student shadow project is an excellent chance for me to try to be  a student of .... for one day. Secondly, it is [what are you talking about here?]a good opportunity to see your teaching ways,[wrong word] atmosphere inside [to experience the atmosphere at ...],  and to deny/prove talk [I don't understand what you mean here] about this university. Thirdly, if I applied for ..... after a year it could be a good motivation for me [to do what?].

 I know that any participation in any project all the time [??]gives any valuable life experience. This is why I always try to be at the end of one’s resources.[I'm sorry, but this doesn't make any sense to me] 

Looking forward to hearing from you soon.[This is very informal.  You might prefer: "I look forward to hearing from you"]

Yours sincerely,