Once Arixmethes becomes a creature (in addition to its other types) due to Part the Waterveil—
Arixmethes will be a Kraken Elemental creature and a land, but no other card types, if it has at least one slumber counter on it (C.R. 205.1a-b, 702.112a, 205.2a, 205.3m), and
Arixmethes will be a Kraken Elemental creature in addition to its other types (but Arixmethes's second ability won't make it a land) if it has no slumber counters on it (note that the awaken effect lasts for the rest of the game, even if the permanent stops being a land) (C.R. 702.112a, 611.2a, 205.1b).
This is because the Part the Waterveil effect has a later timestamp than the effect of Arixmethes's second ability (which began when Arixmethes entered the battlefield), and neither effect depends on the other (C.R. 613.6, 613.6a-c, 613.7).
Note, that Awaken sets the base power/toughness to 0/0. So Arixmethes, Slumbering Isle will be a 6/6 creature due to the 6 +1/+1 counters in either case.
EDIT (Sep. 29, 2020): Correction.
Former Rules Advisor
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