The "easiest" way to see the inscribed area within Vihorlat is to hike up to Sninsky kamen, which is just within the core area, and which offers panoramic views of the overall protected area within Vihorlat. The hike up to Sninsky Kamen starts from Morske Oko, which in turn can be reached by driving to the end of the road from Remetske Hamre (around 20km from Michalovce). Morske Oko seems to be a somewhat popular tourist location (at least for locals) as the site features a car park, signposted trail markers, and there also were several people strolling around the lake. The trail up to Sninsky Kamen, on the other hand, is less popular even if being relatively easy (with the exception of some missing trail markers). In fact, the main challenge is to reach Vihorlat in the first place as the park is located closer to Ukrainian border (only Ukrainian mobile network provider was occasionally within range inside the park) than any major Slovakian city.
In terms of sights, the descriptions of the sights within the other reviews seem rather well fitting also for Vihorlat. The forest areas are rather thick and pristine, with the best areas located close to the peak (and hence the inscribed area). As the area is quite far from large cities and relatively undisturbed by human visitors, the site also offers possibilities for spotting wildlife (managed to spot a fox and a small deer during my visit).
All in all, an area well worth of protection, and a nice (even if not outstanding) location to visit, albeit at a rather remote location.