It is permissible to clean or polish a shofar (ram's horn) on Rosh Hashanah. Many people dip it into wine or vinegar, especially on the inside, to get the sound to come out nice and smooth. All of that is permissible on Rosh Hashanah. It does not constitute metaken mana (repairing a vessel). It's not like fixing a kli (vessel). It's just enhancing something that already works, and therefore it's permissible.
However, if it gets a crack or a puncture, it would be forbidden for a person to fix it on Rosh Hashanah. That's tikkun kli (repairing a vessel). He couldn't even ask a non-Jew to fix it for him, because asking a non-Jew is still asur mi-derabbanan (rabbinically forbidden), and we're not allowed to violate a derabbanan, even in the face of performing the mitzvah de'oraita (Torah commandment) of blowing the shofar.
However, if the thing that you're asking the goy (non-Jew) to do is a mitzvah (commandment), is a issur derabbanan (rabbinic prohibition) in and of itself, and so now you have a double derabbanan (double rabbinic prohibition)—I'm asking the goy, and the goy is doing something which is also derabbanan (rabbinic)—that becomes permissible if it's necessary for the mitzvah. So, for example, if the only shofar they have available is outside the boundary, outside the techum Shabbat (Shabbat boundary), it's permissible to ask a goy to go and bring it—to retrieve it—since techum Shabbat is only derabbanan, and therefore it would constitute a double derabbanan for the purposes of the mitzvah. It would be permissible.