I am not very familiar with web parts.
When I click link “Close” on WebPartZone all this control disappear. How can open again WebPartZone that has been closed before?
I'm assuming you were refering to the WebPart that is being closed from the WebPartZone. If so then basically all closed WebParts are stored in the WebParts PageCatalogPart.. If you wanted to open/add the Webparts that you have closed in the WebZone then you need to use the PageCatalog control.
But can I have just button with click event handler and code in this handler that will open previously closed webpartzone?
You can loop through the WebPartManager WebParts collection and then determine the previous closed WebParts and then use the AddWebPart method of WebPartManager to add WebParts to the WebPartZone: