3 of 3 added 29 characters in body

Mathematica, 253 bytes

f=(For[i=0;j=1,j<=#,j+=Mod[RealDigits[9801/Sqrt@8/Sum[(4j)!(1103+26390j)/(j!)^4/396^(4j),{j,0,125}],10,999][[1,j]]-1,10]+1;++i];"99iiiitS===oooo8862^^^^^^\uusRRRRRRqqqqqqqqMMMMmlllllllgggggggFFFeeeeeeeeDDDDDDaaaaa55555555<<{{\{{{{{{}}}}}}}}--@@@@@@";i)&

Ungolfed:

f = (
   For[i = 0; j = 1, j <= #, 
    j += Mod[
       RealDigits[
          9801/Sqrt@8/
           Sum[(4 j)! (1103 + 26390 j)/(j!)^4/396^(4 j), {j, 0, 125}],
           10, 999][[1, j]] - 1, 10] + 1; ++i];
   "99iiiitS===oooo8862^^^^^^uusRRRRRRqqqqqqqqMMMMmlllllllgggggggFFFeeeeeeeeDDDDDDaaaaa55555555<<{{{{{{{{}}}}}}}}--@@@@@@";
   i
   ) &

Usage is f[999].

A whopping 122 bytes are used to pad the code with a useless string to get the right character frequencies. I'll try to improve that tomorrow.

The character frequencies should match the digits of pi in this order:

3#/&(,![)4+9j0it"S=;]ro862^usRqM1mlgFfeDda5<{}-@

I confirmed that there is some correct order with the following snippet:

Sort[Last /@ 
   Tally[Characters@
     "f=(For[i=0;j=1,j<=#,j+=Mod[RealDigits[9801/Sqrt@8/Sum[(4j)!(\
1103+26390j)/(j!)^4/396^(4j),{j,0,125}],10,999][[1,j]]-1,10]+1;++i];\"\
99iiiitS===oooo8862^^^^^^\
uusRRRRRRqqqqqqqqMMMMmlllllllgggggggFFFeeeeeeeeDDDDDDaaaaa55555555<<{{\
{{{{{{}}}}}}}}--@@@@@@\";i)&"]] == 
 Sort[RealDigits[Pi, 10, 48][[1]] /. 0 -> 10]

I'm computing pi with Ramanujan's series. It converges to 1000 digits in 125 terms. Due to golfing reasons, I recompute the 999 necessary digits for every single digit of the subsequence, but it still completes within a second for n = 999 on my machine.