First off I want to say I don't know anything about the husbandry needs of Schneider's Skinks so going to assume you know what you are doing when it comes to providing the proper husbandry for them.

However, given the fact the previous owner has told you he has been in this condition for a month and for the few days (I assume as you did indicate he would eat a few times for you) and hasn't shown any improvement..leaving a vet trip to the last possible resort is a bad decision. Take him to a vet now.

The fact his eyes are crusted shut is a very clear indication of one of several things. His eyes have been injured or infected and the discharge has sealed them shut. One thought that comes to mind is the previous owner used coiled UVB bulbs which produce harmful levels of UVB radiation (UV Index over 6 is considered dangerous even to us. See www.uvguide.co.uk/phototherapyphosphor.htm about photo-kerato conjunctivitis caused by many brands of coiled UVB compact lights.

I definitely feel now is the time to get him to a vet, instead of waiting till he gets worse. As reptiles tend to hide their illnesses to avoid appearing sickly and weak, by the time it is noticeable usually indicates the problem is getting close to being fatal.
-----
PHLdyPayne