I wondered if there is a way to cut the page in parts and export those in pdf.
Just turn your thinking around bit; export to PDF, and crop in Acrobat. Crop to the pane-1 guide, save a copy as pane-1.pdf, crop to pane-2 guides, save a copy as pane-2.pdf, and so on. You might want to ask if they'll want some overlap. I suspect they will.
Assuming the vertical guides define the panes, I'd suggest you go with John's recommendations.
InDesign itself has an ability to "tile" large-format files into smaller printing panels, but it's designed to match up symmetrically and probably isn't going to get you the results you're looking for easily. His process is probably your best -- and easiest -- shot at getting the output you need.
Thank you! I will work around it then
I hope there is a solution!
ID lets you customize page sizes within the document. So you can have a spread made of different sized pages like this:
Here I've made a 5 page document, selected all of the pages in the Pages panel, and with Allow Selected Spread to Shuffle unchecked, I've dragged the pages together to form a 5 page spread
With Page tool selected in the Tools panel I can select a page and resize and position it via the Transform panel. Here are the 5 pages sized similarly to your layout. I've set the width of page 1 to 4.5"
Page 2 set to 7.125" with its X position at 4.5":
I can then design the spread with crossovers and export the individual pages with bleeds