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I am looking at the following:

The average tallest men live in Netherlands and Montenegro mit $1.83$m=$183$cm. The average shortest men live in Indonesia mit $1.58$m=$158$cm. The standard deviation of the height in Netherlands/Montenegro is $9.7$cm and in Indonesia it is $7.8$cm.

The height of a giant of Indonesia is exactly 2 standard deviations over the average height of an Indonesian. He goes to Netherlands. Which is the part of the Netherlands that are taller than that giant?

I thought to do the following:

Since the height of a giant of Indonesia is exactly 2 standard deviations over the average height of an Indonesian, we get that his height is $158+2\cdot 7.8=173.6$cm, right?

We have the following: enter image description here

Now we want to compute $P(x>173.6)=1-P(x\leq 173.6)$, right?
At the graph we have $173.3$ how could we compute the $P(x\leq 173.6)$ ?

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You are right. $X$ is distributed as $\mathcal N(183, 9.7^2)$. To compute $P(X\leq 173.6)$ you use the standardized radom variable $Z=\frac{X-\mu}{\sigma}$, where $Z\sim \mathcal N(0,1)$

$P(X\leq 173.6)=\Phi\left(\frac{173.6-183}{9.7}\right)\approx\Phi(-0.97)$

$\Phi(z)$ is the cdf of the standard normal distribution. You can look at this table what $\Phi(-0.97)$ is. For orientation, the value is between $14\%$ and $18\%$.

Link to a online calculator.

Let mm be the minimal acceptable height, then $P(x>m)=0,01$, or not? It also equivalent to $P(x≤m)=0.99$, right?

You are right that both equations are equivalent. You have made the right transformations.

$\frac{m-158}{7.8}=2.32 \Rightarrow m=176.174\ cm$ Is this correct?

More or less. We usually say that $\Phi(2.33)=0.99$. It is $\Phi(2.32)=0.98983$ and $\Phi(2.33)=0.99010$. The second value is nearer to 0.9 than the first value.

But the funny thing is that if I use $2.33$ the result is $m=176.174$. Maybe you have used 2.33 on the RHS.

Your answer to the second question is right. $\large \checkmark$

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    How do we know that we have to use the standardized radom variable in this case? – 2017-01-11
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    @MaryStar It is not absolutely necessary to use the standardized random variable. You can calculate $P(X\leq 173.6)$ without out it. But there do not exist a table for X. There are only tables available of the $\color{red}{\text{standard}}$ normal distribution. If you do not standardize the variable you can use an online calculator where you can choose the mean ($183$) and standard deviation ($9.7$). I will post an link to a calculator in my answer. Try it out and double check the result. – 2017-01-11
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    Ah ok. Then to be in the Indonesian basketaball team one has to be at the one percent tallest of the country. Which is the minimum height that someone has to have to be in the team? $$$$ Let $m$ be the minimal acceptable height, then $P(x> m)=0,01$, or not? It also equivalent to $P(x\leq m)=0.99$, right? $$$$ If the Netherlands would have the same minimal height, how many would have height bigger than $m$ ? – 2017-01-11
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    I have done the following: $$P(X>m)=0,01 \Rightarrow 1-P(X>m)=1-0,01 \Rightarrow P(X\leq m)=0.99 \Rightarrow \Phi \left (\frac{m-158}{7.8}\right )=0.99$$ From the table we get $\frac{m-158}{7.8}=2.32 \Rightarrow m=176.174\ cm$. Is this correct? For the second question: $$P(X>176)=1-P(X\leq 176)=1-\Phi \left (\frac{176-183}{9.7}\right )\cong 1-\Phi (-0.72) \Rightarrow P(X>176)=1-0.23576=0.76424$$ Is this correct? – 2017-01-11
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    @MaryStar I have made an edit to answer your questions, – 2017-01-11