Thanks to OP I identified this engine and even found a manual on B&S website. Good job B&S supporting your machinery!
Synthetic oil probably is an overkill, synthetic saves your engine during cold starts. Especially in winter, but unless you use your lawnmower to blow snow ... At 50+ degrees regular oil will do. And synthetic runs longer than regular, I change oil in my cars in 20-25k miles as recommended by Amsoil. Again, we do not need this kind of durability in lawnmowers.
Edit: @earthworm
Use ethanol-free gas, it will not corrode your fuel system during winter.
Ethanol-free gas stations in Pennsylvania