Learn English – A word for a worldly wise person who pretends to be naïve

Someone like you described could be feigning indifference (though this can also be used for a variety of reasons such as feigning indifference to being snubbed, being the subject of gossip, etc.):

feign /feɪn/ vb

1. to put on a show of (a quality or emotion); pretend: to feign innocence

2. (transitive) to make up; invent: to feign an excuse

3. (transitive) to copy; imitate

Etymology: 13th Century: from Old French feindre to pretend, from Latin fingere to form, shape, invent

indifference /ɪnˈdɪfrəns -fərəns/ n

1. the fact or state of being indifferent; lack of care or concern

2. lack of quality; mediocrity

3. lack of importance; insignificance

Absentminded may fit your description:

• tending to forget things or to not notice things : having or showing a lack of attention.

( M-W)