A Just as he got closer to having all of his possessions packed, Hans found it harder to keep his apartment organized because of the increasing number of moving boxes stacked everywhere - IMO "Just as" is redundant
B As he got closer to having all of his possessions packed, Hans found it harder to keep his apartment organized because of the increasing number of moving boxes stacked everywhere - Correct
C As long as he got close to having all of his possessions packed, Hans found it hard to keep his apartment organized because of the number of moving boxes stacked everywhere increasing upward - "increasing upward" redundant
D As soon as he got closer to having all of his possessions packed, Hans found it harder to keep his apartment organized because of the number of moving boxes stacked everywhere increasing upward - same as above
E Just like he got closer to having all of his possessions packed, Hans found it harder to keep his apartment organized because of the increasing number of moving boxes stacked everywhere. - "like" followed by clause is wrong
IMO "just as" is redundant here..
So my choice is B