Mmh, what comes in my mind to replace “little old shop” would be “un charmant petit magasin”, literally translated as “a charming little shop”, “mignon” could also be an option but wouldn’t fit in any sentence.

Here is a book called Un charmant petit village https://www.editionslajouanie.com/un-charmant-petit-village-jean-michel-lecocq.html

There exists an expression : “Ce qui est petit est beau”.

Generally, we only say un petit magasin, un petit village, to indicate something pleasant

PS: When we were young, my sister, much smaller than me, often said this expression ce qui est petit est beau. Useless to detail what she was adding concerning me

I find the following to be a similar expression in both form and meaning: un bon petit …, pronounced un bon p’tit …, and sometimes un bon petit … des familles for extra emphasis.

For example:

On est allés dans un bon petit restaurant.

Je vais me poser tranquille et lire un bon petit bouquin des familles.

It shares with a little old … a connotation of something nice and cozy, easygoing, not extravagant, maybe even a little secret and out of the way.

It’s not a strict equivalent of the English expression though. In particular, it can’t be used for people, unlike "a little old lady", which would be une petite bonne femme.