[RPG] Can a wizard use a 25gp book as a replacement spellbook

Yes, you can cast spells without your spellbook. And yes, your list of prepared spells will stay prepared until you prepare a new list. In the Preparing and Casting spells section, only preparing a new list of spells actually requires your spellbook. Casting just requires spell slots and a prepared list of spells, and regaining spell slots just requires a long rest.

There are some other things you can't do without your spellbook, however:

• You can't cast spells as rituals.
• You can't use your Arcane Recovery feature.
• Most importantly, you probably can't learn new spells. A generous DM might rule that your two new spells learned from levelling up just appear in your spellbook, wherever it might be, but I certainly wouldn't.
• By the same token, I have no idea what would happen with the Spell Mastery and Signature Spells features if you gained them while you didn't have your spellbook. You might not gain them at all; you might gain them once you reclaimed your spellbook.

A suggestion for you: The game assumes that a Wizard has a spellbook, and a lot of the Wizard rules don't quite work without one. If you lose your spellbook, get a new one as soon as possible. For the pittance of 50 gp, you can make the rules work properly again. Even if you still don't change your list of prepared spells, it'll be worth it.

This makes it sounds like a wizard would have a really hard time to prepare a spell from another's spellbook. Is that the case?

That is correct.

As you quoted, each wizard's spellbook notation is unique, therefore you can't prepare spells from someone else's book.

You must first copy the spell into your own book (deciphering the notation in the process), after which you can prepare it as normal.