Is he a brother?
Yes
However, he's not a brother in the sense of a "fellow believer", but merely in the sense of a "fellow man". Let me explain in full:
Word choice
There does seem to be some confusion with the wording here:
James 1:9-10 (ESV)Emphasis added
Let the lowly brother boast in his exaltation, and the rich in his humiliation, because like a flower of the grass he will pass away.
It can be seen that the "lowly" one is a brother. However, it's not clear that the "rich" one is a brother. However, given that these two concepts are presented, in sequence and in the same thought, we can be sure that this "rich" person is a brother.
For comparison, if I said:
Pick up the red ball and the white as well.
Would you think that I was implying anything other than a white ball? It's clear that the intention of the James in this letter was to say that there was a lowly brother and rich brother.
Brother?
While it is clear that the rich man was a "brother", what exactly is being meant here by calling him a brother?
The original Greek word for this is adelphos. Strong's shows that this word was as flexible back then as it is now.
The word could have implied either a physical brother (from the same parent(s)), countryman, a fellow believer, someone having the same national/racial ancestry, etc. The word is quite pliable.
To show how far this word can stretch, another definition (from Strong's) is any fellow or man. Quite simply, the "lowly brother" could have easily just been a "man".
Fellow Believer?
If we take a look at the context of the verse, it seems quite clear that the rich man, while being a "brother", is doomed to perish:
James 1:9-11 (NASB)Emphasis added
9 But the brother of humble circumstances is to glory in his high position; 10 and the rich man is to glory in his humiliation, because like flowering grass he will pass away. 11 For the sun rises with a scorching wind and withers the grass; and its flower falls off and the beauty of its appearance is destroyed; so too the rich man in the midst of his pursuits will fade away.
Summary
The wording here shows that the rich man is a brother. However, this usage of the word "brother" is not meant to imply "fellow believer", but merely another person, such as a countryman or a fellow man. The context of the passage (along with the later reference to James 5) supports this argument.