Can fossil fuels contain energy that came from the sun?

Yes it does. The Sun's energy is used to drive photosynthesis in plants. Carbon dioxide is absorbed from the atmosphere by green plants, and the energy is used to separate carbon and oxygen atoms from the carbon dioxide, a process which absorbs energy. The plant uses the carbon to build tissues like leaves, stem etc., and most of the oxygen is released into the air. This process originally made mammal life possible.

Millions of years later those plants have decomposed and fossilised, and formed oil underground, which is brought up and used as fuel. The carbon in the fuel and oxygen from the atmosphere are recombined by burning the fuel. Re-forming the carbon-oxygen atomic bonds releases the stored energy as heat and generates carbon dioxide. The heat drives an engine and the carbon dioxide is released in the exhaust gases. This is what happens when carbon fuel such as coal, oil and gas are used as fuel.