Yes, these headings appear in Hebrew manuscripts. Here are two of the most famous manuscripts with the heading for your psalm: למנצח אל־תשחת לדוד מכתם בשלח שאול וישמרו את־הבית להמיתו.
- Leningrad Codex — dated to 1008/9 CE — right column, bottom paragraph is the beginning of the psalm.
- Aleppo Codex — dated to ca. 920 CE — right column, bottom paragraph (it's faded, but you can still see enough to find the words)
Among the manuscripts examined by C. D. Ginsburg for his 4-volume edition of Tanakh, none omit the heading for Psalm 59. (I haven't checked every psalm, but there's no reason to expect large-scale omission of Psalm headings.) The headings also appear in our printed (and online) texts, eg. in Mechon Mamre and Sefaria. Note that in Jewish texts, the heading is included in the verses, unlike what is often the case in Christian versions. This is why many psalms are off by one verse when comparing Jewish and Christian versifications.
There are a few headings that appear in old translations (eg. Syriac, Septuagint) that don't appear in the Hebrew editions, but these are isolated examples of disagreements, and not a wholesale rejection of headings as a phenomenon. For example, some Septuagint editions have "for the day before Shabbat, when the earth had been settled" as a heading for Psalm 92 (= 93 in Hebrew editions), which is the daily psalm we generally read on Friday in current Jewish practice; our Hebrew texts do not contain such a heading.