A Closer Look
The Intel Core i5-8500 comes in a standard retail box, which includes a fan heatsink, case-badge, and some literature.
Intel's stock fan heatsink for the LGA115x sockets hasn't changed much within the past decade beyond evolving regulatory compliances (becoming lead-free, RoHS, etc.) It's the same top-flow cooler that has a cylindrical heatsink with radially projecting, forked aluminium ridges, which is ventilated by a 70 mm fan. With its TDP rated at 65 W, you should be able to run the i5-8500 with this cooler.
The Core i5-8500 looks like every other LGA115x processor launched within the past decade. A point to note here is that unlike AMD, Intel is using glue and thermal paste as the interface material between the integrated heatspreader and die. Enthusiasts generally prefer soldered dies. Gamers don't care as long as their machines are quiet enough.
With this generation, the biggest point of confusion has been the package. The 8th generation Core desktop processors bear the "LGA 1151" package markings and look like they'll work on older 100-series and 200-series chipset motherboards. They'll even physically fit on them since nobody at Intel bothered to put the key notches elsewhere. The chips, however, will not work on older motherboards. The machine won't even POST. The box clearly states that you need a 300-series chipset motherboard to use the processor.