What does "About its lot" mean?

Solution 1:

The word lot as intended by Adams in your quote, has a meaning that falls somewhere between those listed as 3a and 3b in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

lot n (bef. 12c) 1 : an object used as a counter in determining a question by chance 2 a : the use of lots as a means of deciding something b : the resulting choice 3 a : something that comes to one upon whom a lot has fallen : SHARE b : one's way of life or worldly fate : FORTUNE

The "lot" of the Monk's faith in a particular thing is in one sense the share that comes to that faith in his active belief system, and in another sense it is the fate of that faith as a living idea in the Monk's mind; in either case, it has a short shelf-life.

Adams's phrase "twenty-four hours was usually about its lot" might perhaps have been clearer if he had worded it as "twenty-four hours was its usual lot"—that is, 24 hours was the typical lifespan of one of the Monk's unshakable (but somehow ultimately discarded) beliefs.

Solution 2:

It means as much as one can sensibly manage or sufficient.

Probably derived from either

Collins English Dictionary http://www.collinsdictionary.com/dictionary/english/lot
3. portion in life; destiny; fortune
6. an assigned or apportioned share
but probably closer to this definition (13) which is NOT in Collins online but is listed here http://www.thefreedictionary.com/Lot as belonging to Collins
13. the lot, the entire amount or number

Some quotes

Bare Knuckle Bitch by Marcus Blakeston 2012 Google Books
Just one drink though, and that's my lot. After that it's straight home.

Just for Kicks by Kenny Logan 2009 Google Books
'Another £100,' I said, 'and that's my lot.' James was loads up.

You'll notice both those quotes are my lot and not it's lot but it's the same principle. One could say

My car can just about do a hundred, but that's it's lot.
or with their
I'll give them another thousand dollars, but that's their lot.

So in your original quote about it's lot could be replaced by about as much as the monk could manage.