Is there a problem to solder only one side of a dual PCB?
Yes, there is a problem. You won't have reliable connections from the top tracks to the components unless the holes are plated through - which seems unlikely as you are making the boards yourself.

If you haven't made the board yet you could:


Make a single sided board and use wire jumpers to bridge across tracks where needed. Much simpler to make and all soldering is on one side.
If you really want to try a double-sided board then use pins (Vero used to make these in my youth) or wire links to connect through the board. The pins would have to be soldered both sides of the board but all the components only on the bottom. So, if blue tracks are on the top side, the PIC pin 1 would need a short track on the bottom side to a through hole to the top side to connect to the blue track somewhere other than at pin 1 itself.