This is not an exact answer, but it helps to quickly determine in many cases that a given function is primitive recursive. The idea is to use a reasonable programming language in which your function can be expressed more easily than with "raw" arithmetic and primitive recursion.

Suppose a function $f : \mathbb{N} \to \mathbb{N}$ that is computed by a simple imperative program. More precisely, the program is allowed to use basic arithmetic operations (but no funny stuff like the Ackermann functions), variables, arrays, loops, and conditional statements. Further suppose that the running time of computing $f(n)$ is polynomially bounded by $n$ (I am simplifying here, we could also deal with more complicated bounds and more complicated languages). Then the function $f$ is primitive recursive.

Let us apply this to your question. The following Python program computes the function $\pi(n)$, uses just a couple of loops, basic arithmetic (we could replace the remainder function % with another loop), and has running time quadratic $n$, assuming all basic operations are constant time:

    def pi(n):
       '''Computes the number of primes which are less than or equal n.'''
       p = 0 # the number of primes found
       k = 2 # for each k we test whether it is prime
       while k <= n:
           j = 1 # possible divisors of k
           d = 0 # number of divisors of k found
           while j <= n:
               if k % j == 0: d = d + 1
               j = j + 1
           if d == 2: p = p + 1
           k = k + 1
       return p

Therefore your function is primitive recursive.