What is the optimum way to store, update, and analyze integer data for 100+ attributes?
Assumed business rules
As I currently understand your specifications, the following assertions are of prime relevance:

A User inscribes zero-to-many Texts.
A User registers zero-to-many Characters.
A Character is counted in zero-to-many Sums.
A Text contextualizes one-to-many Sums.
A User computes zero-to-many Sums.

Logical data model
And then, from the above enumerated assertions, I have derived the IDEF1X logical data model shown in Figure 1.

As you can see, with this structure you could have a series of Sums related to a specific Character in the context of a particular Text. Instead of performing an UPDATE each time that a new Sum is computed, you just have to INSERT a new row, which would hold the exact point in time that such Sum was calculated.
In this way, you do not need to have a table with about 100 columns, instead, as stated in comments, I suggest using a Character table that stores each character occurrence in an individual row.
I have also added a User entity in order to depict a more complete scenario, since I assume that all the sums (or counts) are computed by a determined user, probably with the aid of an automatic method.
This structure allows you to analyze with ease the trends of the sums that a given Character presents in relation to a particular Text by comparing the set of values contained in the aforementioned Sum series.