Can I install a modern light fixture on this box?
I would install a new led flat panel fixture. No new box or patching needed. 

Remove the threaded stud so the panel can sit flush to the ceiling. 

These lights do not weigh very much and you can attach them to the ceiling instead of the box if you use an appropriate anchor or if you are lucky enough to have some wood structure in the ceiling line up with the mounting holes of the fixture. 

I have installed these by myself but having an extra set of hands is very helpful. 
On the ones that i have installed the cover comes off, you then hold the main body of the fixture up to the ceiling and use a pencil to mark the location of the mounting holes. If there is structure in ceiling that falls within the parameters of the fixture but is not where the mounting holes are you can drill new holes in the fixture if there is nothing restricting access and the new location does not interfere with the cover seating properly. 

Your ceiling seams to have had second layer of drywall added so finding structure may be a challenge but that means that an anchor will possibly have more holding power. Depending of the size of the fixture you may want to secure it with anchors at all four corners but two may be sufficient.