Hey my wizard just found an arcane grimoire (+1). He doesn't have enough money to copy the spells in the new book, but will he get the bonus to attack, saving throws and arcane recovery if he just uses the book as spellcastin focus even if it isn't his main spellbook?  

Quote from nonemous48086 >>

thats not how this works

 
Actually, it is. The bonus applies "to your wizard spells" when using the book as a focus. It is not limited to the spells in the book. It is to any Wizard spell you may cast.

Quote from nonemous48086 >>

the part it doesnt work like is writing spells off of it. It doesnt have spells in it for u to copy

 
Well, no, you're supposed to add them to your character like any other spell you scribe. 

Quote from nonemous48086 >>

the part it doesnt work like is writing spells off of it. It doesnt have spells in it for u to copy

 The OP was talking about adding spells to the Arcane Grimoire from their original spellbook, since the Grimoire can also function as a spellbook.

you're super fun at parties

Assuming you have attuned to the Arcane Grimoire, you may use it as an arcane focus for casting any spell you have prepared, and receive the +1 bonus to your spell attack/spell save DC.  The item description in Tasha’s does not imply that this item is generally found containing spells, although there is no reason it couldn’t.  In that case, I suppose you would have to follow the standard rules for copying spells from a found spellbook into your primary spellbook.
Regarding money/time, if you want to use this item as your new primary spellbook, remember, it only costs/takes 10GP/1 hour per spell level to copy spells out of your OWN spellbook into a new one.

If you are an order of scribes, you can summon the spirit of your old spellbook into the arcane grimoire and add all of your old spells to it, therebye turning it into your new spellbook. to answer your actual question, yes, RAW it will apply to spells not in the spellbook