It should read:
I can recall vivid images of my first atlas; me, sitting at a table, finding all countries and their capitals on the atlas, and jotting them down on a paper.
'Me' is, as per the OED:
The objective case of the first person pronoun I, reproducing the Old English accusative and dative.
The important part is sense 5h:
For the subjective pronoun I. (h: Introducing a sentence or clause with a participle.)
Where the pronoun 'me' introduces the following clause. A relevant example provided is the 1950 one which I have included in the quotation. I'm sure you can see the similarity.
1950 J. Hersey Peggety's Parcel of Shortcomings in Fling (1990) 22
"Me, sitting there, taking up half the back seat of the car."
The semicolon here is used either instead of a conjunction, or to link two independent, closely related sentences, and it may make more sense to you if I remove the semicolon and replace it with a conjunction. e.g:
I can recall vivid images of my first atlas with me, sitting at a table, finding all countries and their capitals on the atlas, and jotting them down on a paper.
or,
I can recall vivid images of my first atlas. Me, sitting at a table, finding all countries and their capitals on the atlas, and jotting them down on a paper.