The answer is 2261, isn't it?

Since $x$ is neither a prime number nor a perfect square, we can have only $6$ or $8$ as the first digit.

The difference between the number and $100x$ would obviously be a $2$ -digit number. There are $4$ perfect cubes under $100 - 1, 8, 27, 64$

So we have the following numbers :

$601, 608, 627, 664 \\ 801, 808, 827, 864$

Now, the $3^{rd}$ condition is followed by only one number, $\boxed {827}$ , and were $\text{DONE}$ !

Upvote it if you liked it

Python 2.7:

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# non-prime and
# non-square digits
possible_x = [6,8]
def cube(n):
    test = 0
    while test**3 <= n:
        if test**3 == n:
            return True
        test += 1
    return False
def palindrome(n):
    return str(n) == str(n)[::-1]
def goal(guess):
    x = int(str(guess)[0])
    if x not in possible_x:
        return False
    if not cube(abs(guess-100*x)):
        return False
    if palindrome(guess-100):
        return True
for guess in xrange(100,1000):
    if goal(guess):
        print "Answer:", guess