It is not exactly the same structure as "gustar". The "le" in your sentence is a dative of interest. The verb "acaban" is in third person plural because the subject is an unspecified third person, as explained here. The translation of the sentence
A nuestro vecino le acaban de instalar aire acondicionado en su casa
- They have just installed air conditioning in our neighbor's house
or more naturally
- Our neighbor has just had air conditioning installed in his house.
The indirect object pronoun le means roughly for him, as in "they installed the air conditioning for him". The third person plural "acaban" is because the subject is unspecified and irrelevant. For such unspecified subjects, the plural is used, even if the subject may actually be a single person. A similar example is
- Le han cortado el pelo muy mal (They have given him a very bad haircut)
where the hairdresser is probably a single person, but still the third person plural han cortado is used.
The sentence you suggest in 2.
Nuestro vecino acaba de instalar aire acondicionado en su casa
has a different meaning, because it implies that the neighbor has installed the air conditioning himself.
Additional note: the title of your question mentions a "prepositional phrase", though the body seems to ask about something else. To clear any doubts, "acabar de + verb" (see meaning 11) is a standard verb phrase that refers to an action that has just been completed. The translation could be "have just + verb".