Should I speak to this person privately and advise him?
Or should I tell the boss above me about him, and let whatever will happen happen?
Or should I prevent him from using the Internet, and in this case what excuse should I give for preventing him?.
If it is proven that this employee is going to permissive websites, then you must advise him and warn him, call him (to proper Islamic behaviour) and guide him, and remind him of the negative consequences and sins that result from going to these websites. If he responds, then praise be to Allaah, but if he persists in that, then there is nothing wrong with telling the director of the centre, so that he can take suitable measures concerning him. The scholars permitted mentioning a man’s faults and bad deeds for the sake of seeking help to change evil, if it is not possible to change it otherwise.
Al-Nawawi (may Allaah have mercy on him) said in his commentary on Saheeh Muslim: But it is permissible to speak of a person in his absence for a legitimate shar’i purpose, such as seeking help to change evil and bring a sinner back to the right path. So he may say to the one whom he hopes is able to help: So and so is doing such and such, so deter him from that, and so on … end quote.
But that should only come after advising him and exhorting him in private.
May Allaah help us and you to obey Him and seek His pleasure.
And Allaah knows best.