This is a parlor-style sentimental waltz typical of early 20th-century popular sheet music. It is distinguished by its charming, simple melodic contour and nostalgic aesthetic.
The primary challenge is maintaining a steady, graceful waltz rhythm while balancing the accompaniment with the vocal-style melody.
Comparable difficulty and character — good candidates for what to learn next.
'Neath the Old Cherry Tree, Sweet Marie is rated Henle level 2 of 9 — beginner repertoire, roughly ABRSM grades 2–3. In the RCM system it corresponds to about level 3.
You should be comfortable at Henle level 2 — meaning you can already play level 1–2 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of 'Neath the Old Cherry Tree, Sweet Marie lasts about 3 minutes.
'Neath the Old Cherry Tree, Sweet Marie is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Blarney by Egbert Van Alstyne, 'O sole mio by Eduardo Di Capua, Believe Me, If All those Endearing Young Charms by Irish Folk Music — see the full similar-pieces list on this page.