In (1) we have a relative clause after the noun "services," and that renders the definite article necessary, as customers are not said to have to pay more for all additional services, but only for the ones they request for their computers. If that defining relative clause were not there and the additional services had not been defined in the context, we'd find the zero article:
(1a) Customers need to pay more for additional services. (What those services are, we just don't know.)
I don't find (2) to be a correct sentence, mainly because of the first occurrence of the preposition "for." I'd use "about": one informs or lets others know about something. In this case, since information needs to be provided about something unknown and unspecified, I'd use "any" rather than the definite or the zero article, and -- although it is not strictly necessary -- I'd place "for your stay in our hotel" within a relative clause that points to the future time, in line with the adverbial "in advance":
(2a) You need to let us know in advance about any additional services you will/may require for/during your stay in our hotel.