Honda Accord Sport 2018, OEM shocks and After market Lower Springs?

I'm thinking of doing a 1 inch drop with the H&R Sport Front and Rear Lowering Coil Springs, they go for about $250

Reason being I plan to add new wheels
20 by 9, which lowering is a must for new wheels in order to give it that sexy and more aggressive look to the car.

Will my shocks run into problems?

Yep, could be too 'long' when compressed all the way…

In a way you have no choice because no one makes performance shock/strut for it or that H&R offers a cup kit. In general one inch should be ok even with the different spring rate and trip of the lowered springs.