Real word, warm feeling — but it's an adjective you exclaim, not a noun you wear.

What a Japanese person actually thinks
Slightly odd on a body, because it's an exclamation. It reads a bit like tattooing 'HOW LOVELY!' — a response floating without the thing it's responding to.

Meaning

Literally: bosom/cherish + adjective ending
Actually: The warm rush of fondness when something reminds you of the past. Crucially it's a reaction: you *say* 懐かしい when you see an old photo.

Scores

Naturalness●●●●●

Extremely common word. The register mismatch is with tattoos, not with Japanese.

Coolness●●●●●

Five characters, three of them hiragana. Visually loose and grammatically incomplete as a standalone statement.

Watch out

lowUnlike English 'nostalgia', 懐かしい is warm and pleasant — there's no ache of loss in it. If you wanted the ache, that's 郷愁 or 物の哀れ.

Get this instead

郷愁 kyōshū
homesickness; longing for a lost place

The noun, with the ache included. Two dense kanji, no hiragana.
追憶 tsuioku
recollection; dwelling on the past

Memory as an act rather than a reaction. Literary and strong.

On skin

懐かしい

Orientation: vertical

Brush.