Your rewriting doesn't really work, sorry. 'At least' means 'a minimum of'. It is better read as (arise) (at least in part) from.... They are saying that having to adapt(...) leads to problems. They are part of the cause, and probably more that just 'a part', hence the 'at least in part'.

"Arise at least" actually goes with the next part of the sentence: "arise at least in part from". "At least in part" is a phrase that means "at least a few of the things involved". So, the sentence is saying that many problems might face small businesses but "having to adapt to rapid advances in technology" is at least one of them!