Without object, the sentence cannot be made passive.

The idea of making anything in passive is to interchange the subject and object and thus make the sentence more meaningful or to reveal/hide the identity. Consider these reasons for using the passive voice.

Do you want to play? lacks object in that

Add some object and it'll be possible.

1

I think the reason the book gives the answer stated in #1 is the subject of your question: passive construction of infinitive verbs. (I can't be sure because I don't have your book in front of me.) There are two verbs in the original sentence: "expects" and "to help." If the book asks you to convert the infinitive to the passive, then you need to change "to help" to "to be helped (by X)".

Note 1: "It is expected" is the proper way to make the passive form of "expect", but for whatever reason "It is expected by him" is rarely used. If you do use it, sounds better to my American ears to say "It is expected by him to be helped by me." However, the book answer is more natural at least in my northwestern American dialect.

Note 2: "It is expected that" is also very rarely used, and usually only in reference to a future event. If you do use a that-clause in this case you need to drop the "should" because it actually becomes a subjunctive construction: "It is expected by him that I help him." However, this should probably be avoided because this construction is so rarely seen.

2

"It is time for X" is simply the common idiom when the passive voice is used. Again, if you use that you need to drop the "should" and write "It is time that action be taken." However, it's far more common to see "it is time that" with a subject directly after, such as in "It is time that we take action." This wouldn't answer the book question correctly, though, since it asks for a passive construction.