You need to restart the process of acquiring ritual spells via this mechanic.
The process of copying the spell into your ritual book takes 2 hours
per level of the spell, and costs 50 gp per level.
The feat mentions this.
When you choose this feat, you acquire a ritual book holding two
1st-level spells of your choice
This occurs only when you choose the feat.
Afterwards the content of the book, a physical object, is governed by the process described above. So if you lose it, your only recourse is to find more spells and use the rule above to scribe rituals into a new book.
As a general comment, the gist of the game is for the referee to present a setting for the player to experience as their character. In this sense it is a pen & paper virtual reality. This relates to your question in that given the lack of specific rules, you should consider the issue as if you were really there. The Warlock getting his Book of Shadows (PHB 108) is an example of a specific exception to the assumption of a pen & paper virtual reality. In this case the character has a physical book filled with ritual spells. Lose the book and you lose the spells and have to spend time and money to make a new one.
Note that there appears to be no initial cost assigned to making a new ritual book. However, in the equipment list a spellbooks cost 50 gp, and a blank book 25 gp. Personally, I would charge a PC the spellbook cost for a new ritual book. However, if you are not comfortable with that then charge the player the cost of a book. Other than that, there is nothing to prohibit characters from spending additional time and money to make a backup book.
Also, this is consistent with how the Wizard spell book works on page 114, which is the closet relevant mechanic.
Of interest is that the ritual feat is the only way for a sorcerer to cast ritual spells, as he doesn't possess a ritual casting feature as a function of class as far as I can tell.
Best Answer
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.
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.
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.