NotMore than a few people think that reading books is not one ofjust a hobbiesy, but a cultural heritage we should pass on to our children.

「少なくない」と一番近いの英語表現なら、多分「More than a few」だと思います。他には、「Quite a few」でも結構近い。

Pass to => Pass on to
ここは「on」を必要となります。
Pass to =　渡る
Pass on to = 継ぐ、伝える
明確に説明できませんですが、「on」に付けると、自分に経由で渡るの意味ではなく、自分に受けたものを次の人等に継ぐの意味になります。

Because I am such a persomeone who cannot do without books, I would do anything to spread the joy of reading.

I mean the moreThe busier we get busy, the less time we can spend on them.

Not a fewSome people think that reading books is not one ofjust a hobbiesy, but a way to pass cultural heritage we should pass to our children.

Because I am such a person who cannot dolive without books, I would do anything to spread the joy of reading.

Today, it seems enviable that youthat one who can spend yourone's free time on books to be envied.

"It seems enviable" is incorrect: people are enviable (and we envy other people), not so much situations and things.

I mean the moreat the busier we get busy, the less time we can spend on them.

"That" seems preferable here, to indicate that you're setting off a clause, as the wording is not the typical subject-predicate structure. "more" + "busy" = "busier"

Not a few people think that reading books is not one ofa hobbiesy but rather a cultural heritage we should pass to our children.

"Rather" seems appropriate here to mark off the contrast. "Not a few" is possibly awkward but acceptable.

Because I am such a person who cannot do without books, I would do anything to spread the joy of reading.

Or "because I am such a person, I would do anything." I think you could also keep it as it, but you would have to adjust the phrasing.

Today, it seems enviable that you can spend your free time on booksspending your free time reading books is invaluable.

"on books" sounds awkward here. "Reading books" is easier to understand

I mean the more we get busy, the less we can spend on thembusier we are the less time we spend reading.

if we say "...spend on them" it sound awkward again because we are implying once again "on books" which we don't really say

Not a fewMany people think that reading books is not one ofly a hobbiesy but a cultural heritage we should pass to our children.

I think I know what you mean to say here. We use a "...not only....but..." structure. Like "...not only is she nice but she's smart"

Because I am such a person who cannot dolive without books, I would do anything to spread the joy of reading.

I used "live" so that you don't have two "do's" in your sentence.