How many New Testament Manuscripts do we have from the period between the first century and the fourth century? How many early New Testament Manuscripts do we have from the first four centuries of the Christian faith? 
(It does not have to be a complete manuscript of a book to count)
How many early New Testament manuscripts do we have?

You ask about the first four centuries, until about 400 AD. I am inclined to initially answer through 300 AD, because I believe that period is more important for historical* reasons.

This page on Wikipedia lists the 130 papyri of the New Testament that we know of, together with what content each has. For our purposes, we need to sort each of the three tables by date, then count the early ones. Be careful counting, because the same papyrus may appear multiple times in the table.

I count 27 in the first table, 19 in the second, and 22 in the third. The total number of NT papyri in the first three centuries is then about 68.

Very few codices (the other types of manuscripts) were written by AD 300 as far as I know, so that should be close to the total count for all manuscripts.

Another web page has a similar table, but in date order. The result was different, but in the ballpark. On that site, I count:

Manuscripts up to 300 AD: 56 (53 papyri and 3 uncials)

Manuscripts up to 400 AD: 102 (the "versions" are not in Greek)

*About 300 AD, the Emperor Diocletian severely persecuted the Christian church, and probably destroyed a large percentage of the Bible manuscripts extant at that time. There have been disputes whether the best or worst manuscripts were destroyed, and how the Church chose to replace them later. Apparently there was a single order for fifty copies of the Bible later, paid for by the Emperor Constantine if I remember right. Some have suggested changes were deliberately made in those copies.