You want $\sum_{n = N+1}^{\infty} \frac{1}{n^4} < .005$.

We have the integral comparison $$\int_{N+1}^{\infty} \frac{1}{x^4}< \sum_{n = N+1}^{\infty} \frac{1}{n^4}< \int_{N}^{\infty} \frac{1}{x^4}$$ or

$$\frac{1}{3(N+1)^3} < \sum_{n = N+1}^{\infty} \frac{1}{n^4} < \frac{1}{3N^3}$$

The equation $\frac{1}{3N^3} = 0.005$ has the solution $N = 4.05...$ and we see that

$$\frac{1}{3 \cdot 5^3} < \sum_{n = 5}^{\infty} \frac{1}{n^4} < \frac{1}{3\cdot 4^3}$$

or

$$ 0.00266 < \sum_{n = 5}^{\infty} \frac{1}{n^4} < 0.00520$$

Still that does not guarantee that the remainder starting with the fifth term is less than $0.005$ but we are close. Let's get a better approximation:

$$\sum_{n = 5}^{\infty} \frac{1}{n^4} = \frac{1}{5^4} + \sum_{n = 6}^{\infty} \frac{1}{n^4}< \frac{1}{5^4} + \frac{1}{3\cdot 5^3} = 0.0016+ 0.00266... = 0.00426... < 0.005$$

Thus the first such $N$ is $4$ ( anything larger OK too).

$\bf{Added:}$

Thanks to the observation of @Claude Leibovici: we check

$$\frac{\pi^4}{90} - ( 1+ \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4}) =0.00357...$$ while

$$\frac{\pi^4}{90} - ( 1+ \frac{1}{2^4} + \frac{1}{3^4} ) = 0.00747...$$