Hello Lea,
there is only one Axel or Arthur Berg on the emigrant CD which fits the parameters you gave; i.e. born ca 1871, emigrated ca 1881.
This is Axel Berg, age 9, emigrated from Söderala parish, left the port of Malmö on 25 May 1882 with destination St Paul, MN.
He travelled with his parents and brother. They were Sven Berg, age 41, sawyer by profession, Anna Berg, age 39, Carl Berg, age 11.
Could this be the right person?
Does your great grandfather?s death certificate give his parents? name?