No, spells from the Ritual Caster feat do not have to be prepared by any class.
You can confirm this by looking at the general rules for casting ritual spells alongside the specific text of the Ritual Caster Feat.
Certain spells have a special tag: ritual. Such a spell
can be cast following the normal rules for spellcasting,
or the spell can be cast as a ritual. [...]
To cast a spell as a ritual, a spellcaster must have a feature that
grants the ability to do so. The cleric and the druid, for example,
have such a feature. The caster must also have the spell prepared or
on his or her list of spells known, unless the character’s ritual
feature specifies otherwise, as the wizard’s does. (PHB pg. 201-202)
Now let's look at the first few lines of Ritual Caster, since the rest mostly just relates to your spellcasting stat and adding new spells to your ritual book.
You have learned a number of spells that you can cast as
rituals. These spells are written in a ritual book, which
you must have in hand while casting one of them. (PHB pg. 169)
If I am interpreting your question correctly, your confusion stems (at least in part) from that last line of the ritual rules, "The caster must also have the spell prepared or on his or her list of spells known, unless the character's ritual feature specifies otherwise, as the wizard's does."
However, keep in mind that nowhere in the Ritual Caster feat is there discussion of preparing spells. Indeed, insofar as preparing spells is a feature of certain spellcasting classes, Ritual Caster doesn't convey that feature. What it does give you is the ability to cast spells as rituals (provided they are in your ritual book and you are holding the ritual book) and a couple of spells to start off with in your ritual book.
As HellSaint pointed out, if a caster who normally has to prepare spells takes the Ritual Caster feat, they still do not have to prepare the spells in their ritual book in order to cast them as rituals. In fact, if they don't know the spell outside of the context of the feat they probably can't prepare it.
Though the spells in your ritual book do not cleanly fit the criteria of "prepared or on his or her list of spells known", I think the Ritual Caster feat does clearly otherwise specify the criteria for spells you can cast as rituals.
tl;dr - The spells you gain access to from the Ritual Caster feat are, by and large, totally separate from any other spellcasting ability you may have. You can cast those spells as rituals because they are in your ritual tome and you are using the tome as the feat describes. Your class does not even come into it.