None of the original manuscripts (autographs) of the New Testament (or Old Testament) are extant.
Manuscripts
The oldest known New Testament manuscript is P52, a small-fragment of the Gospel of John (in Greek), often dated to about AD 125. Depending on the date you subscribe to for the authorship of John, that would put this copy about 25-60 years after the original was composed. For an ancient document, this is an insanely early copy.
By way of comparison, the earliest surviving copy of Tacitus is almost a millennium after he wrote it (see here), and the earliest surviving copy of Esther may be 1500 years after it was composed.
A chart of the earliest surviving copy/fragment of each book of the New Testament can be found in the Earliest Extant Manuscripts section here.
The earliest surviving complete New Testament is Codex Sinaiticus (see source above), from the 4th century.
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Language
Most/all of the New Testament was originally written in Greek, and so efforts to determine what the original text said--or to get as close as we possibly can to what it said--prioritize the Greek over other languages. For some background on the relevant textual criticism, see my post here on BHSE.
There are historical arguments, linguistic arguments and possibly a surviving manuscript suggesting the Gospel of Matthew could have been originally composed in Hebrew. The aforementioned surviving Hebrew manuscript, however, is not nearly as old as the oldest Greek copies of Matthew, and is at best a severely corrupted descendant of a Hebrew original.
There are outlier arguments that Hebrews may have been originally composed in Hebrew or Mark in Latin, but generally the only New Testament document where there's any significant debate against the view that Greek is the original language is the Gospel of Matthew.
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Conclusion
Greek will get us closer to the original wording than New Testament manuscripts in any other language.
