Suppose sufficient balls are placed at random in n boxes up until the
first point at which only x boxes are empty, and that number of balls
is r: then the probability that exactly r balls are used for given x
and n is
S2(r-1,n-x-1) * n!/(x!*n^r)

where S2(a,b) are Stirling numbers of the second kind and can be
calculated with 
S2(a,b) = b*S2(a-1,b) + S2(a-1,b-1) starting with 
S2(0,0)=1 and S2(a,0)=0 if a>0.

In this case x=0 and n=36 so the probability becomes
S2(r-1,35) * 36!/36^r

This is not difficult to calculate for individual values of r except
that some of the numbers are very large.  

As far as I can tell, the median is r=142, the 99% cumulative
probability comes with r=291, and the mode is at r=129.

These compare with the easier to calculate mean of  
Sum_{1<=j<=36} 36/j = 150.284...
[ in general Sum_{x+1<=j<=n} n/j ] 
and standard deviation of
sqrt(Sum_{1<=j<=36} (36/j)*(36/j-1)) = 44.114...
[ in general sqrt(Sum_{x+1<=j<=n} (n/j)*(n/j-1)) ] 
so the 99% point comes almost 3.2 standard deviations away from the
mean, which is plausible.

As far as I can tell ... the 99% cumulative probability comes with r=291

A basic result:

Suppose r balls are placed in n boxes at random and x boxes are empty,
then the probability that x boxes are empty for given r and n is 
Prob(X=x|r,n,C) = S2(r,n-x) * n! / (x!*n^r) 

Proof by induction: 

a) True for r=0 since
if x=n: Prob(X=n|r,n,C) = 1 = S2(0,0) * n! / (n!*n^0) and 
 if x<n: Prob(X=x|r,n,C) = 0 = S2(0,n-x) * n! / (x!*n^0) if x<n

b) Assume true for r=t-1 then
Prob(X=x|t,n,C) 
= Prob(X=x|t-1,n,C)*(n-x)/n + Prob(X=x+1|t-1,n,C)*(x+1)/n
= (n-x)*S2(t-1,n-x)*n!/(x!*n^t)+(x+1)*S2(t-1,n-x-1)*n!/((x+1)!*n^t)
= ((n-x)*S2(t-1,n-x)+S2(t-1,n-x-1))*n!/(x!*n^t)
= S2(t,n-x)*n!/(x!*n^t) since S2(a,b)=b*S2(a-1,b)+S2(a-1,b-1)

c) so by induction the basic result is true.

The result I started with is simply saying that 
if R is the the first number of balls at which X=x then  
Prob(R=r|x,n,C)
=Prob(X=x+1|r-1,n,C)*(x+1)/n
=S2(r-1,n-x-1) * n! / ((x+1)!*n^(r-1)) * (x+1)/n
=S2(r-1,n-x-1) * n! / (x!*n^r)