Formulate the relevant hypotheses and test statistic (including its distribution) 
A manufacturer of space shuttle light bulbs claims that the defect rate of the bulbs is $0.1\%$. You suspect the defect rate is actually higher, so you have checked $1000$ identical light bulbs from this manufacturer and found out that 3 of them are defect. 
  Formulate the relevant hypotheses and test statistic (including its distribution) and investigate the claim of the manufacturer at significance level $α = 0.05$.

I have problems to figure out which distribution it is, in my opinion, is a Poisson distribution but I'm not sure, can someone confirm it? 
For the hypothesis, it's correct to assume that $H_0:\mu=1$, $H_1:\mu>1$? 
Then how can I use the significance level with a Poisson distribution? (I know how to solve it but with a Normal) :(
Edit: 
$P(X\ge3)=1-P(X<3)=1-(P(X=0)+P(X=1)+P(X=2))$ 
For a $Bin(n,p)$ $P(X=k)=\frac{n!}{k!(n-k)!}\cdot p^k\cdot(1-p)^{n-k}$ 
In order to calculate the probability, I have to substitute the values for $n$ and $p$ in the above equation and for each case the values of $k$ from $0$ to $2$. 
Edit 2: 
$P(X\ge3)=1-P(X<3)=1-(P(X=0)+P(X=1)+P(X=2))= 1-0.981=0.019$ (from calculator), then how can I formulate a relevant hypothesis?
 A: Hint:
@lulu is correct that this is answered most directly by using the binomial distribution. Here is output (slightly edited for
relevance) from Minitab's exact
binomial test procedure. 
Because this seems to be a homework problem, I will leave it to you
to interpret the P-value and to work out how it was
obtained from the distribution $\mathsf{Binom}(n=1000,\,p=0.001).$
Test for One Proportion 

Test of p = 0.001 vs p > 0.001

                                Exact
Sample  X     N   Sample p    P-Value
1       3  1000   0.003000      0.080


Note: Your idea to use the Poisson approximation to the binomial distribution is not wrong. However, I suppose the exercise intends for you to use binomial. 
The Poisson distribution that approximates $\mathsf{Binom}(n=1000,\,p=0.001)$ is $\mathsf{Pois}(\mu = 1).$ (The approximation is very good.) If $X$ has this Poisson distribution, you would expect to see one defective bulb: $E(X)=1.$ But you have seen three defective bulbs. That is more than expected; the question is whether it is enough more than expected to be called 'statistically significant' at the 5% level. Can you find $P(X \ge 3)?$
The plot below compares the PDFs of $\mathsf{Binom}(n=1000,\,p=0.001)$ and $\mathsf{Pois}(\mu = 1).$ 

Addendum: From R: x = 0:4;  pois.pdf = dpois(x,1);  bino.pdf = dbinom(x, 1000, .001); 
cbind(x, pois.pdf, bino.pdf) returns (ignore line numbers in brackets):
     x   pois.pdf   bino.pdf
[1,] 0 0.36787944 0.36769542
[2,] 1 0.36787944 0.36806349
[3,] 2 0.18393972 0.18403174
[4,] 3 0.06131324 0.06128251
[5,] 4 0.01532831 0.01528996

