How do I display menu in header without loosing styles?

In order to convert a plain html template into a working WordPress-theme you need to do more than just “a couple of steps”.

To answer your question:

Option 1)

Custom Menu Walker

Besides the work of converting the rest of the template, you need to write your custom menu walker class in order to meet the mark-up of given template. That is the PHP-class, which will render your menu in the frontend.

Option 2)

Edit the styles (css) of the template

The other way around would be to use the default markup of the menu-output and just edit the CSS of the template, which you have purchased. That is less complicated compared to option 1.

See:

WP Nav walker Core docs

Tutorial Walker Class

I have been working for more than six years with WordPress now and even after that amount of time I still don’t know all the functions and possibilites of it.

So, my/the bottom line is: you should mentally prepare that this will be a long trip, learning WP. Learn WordPress templating first.

The good news: there are tutorials all over the internet about every thinkable topic.

Start by having a look at the core page for theme development