If I search for a product name on Bing, for example "paint.net", the first result is something that looks like an advertising area, but it's called "Official Site".
How is it possible to get something like that for my own products?
At least not by forcing it. The feature is called Best Match, and it's driven algorithmically.
You need to sign up to Microsofts AdCenter.
Like Google Adwords, AdCenter is based on a Pay Per Click Advertising model where you pay only when they click on the link and visit your site.
The Cost Per Click (CPC) is based on a variety of factors including but certainly not limited to: relevancy, Click Through Rate (CTR), speed of landing page.
More information is available here: Advertise on Bing using Microsoft adCenter