I'm presuming here that each player's primary goal is to win (i.e. claim, or otherwise be rewarded with) as much land as possible, and that a secondary, tie-breaking goal is to minimize the amount of land won by the other player. I'm also assuming that by "contiguous border" you mean that each region must be defined by a single and non-self-intersecting contour, which means that in addition to each region being contiguous, donut-shaped regions are not allowed. (One couldn't construct a donut-shaped region by extending a C-shaped region until its points touched, as the shared border point would then not be a "border" at all.)
Observe that the first region is limited in a way that no later region is limited: It cannot separate the unclaimed land into multiple contiguous areas, because of the contiguous border condition. WLOG, assume that this first region is a circular-ish blob.
In contrast, B's first region can split the remaining unclaimed land into as many distinct areas as B likes. An example region would be a star-shaped region, whose center is the antipode to the first placed regions, and all of whose outer points reach around the globe to touch the first region. An N-pointed star will split the remaining area into N regions. N can then be chosen such that all regions are below $10^3 m^2$.
If B chose to do this, they would get A's first region (as a reward), and their own; the remainder of the world would presumably become an exceedingly annoying set of fantastically long and thin wildlife preserves. This would be the optimal strategy if the players' primary and secondary objectives were reversed.
B's other, friendlier option is to pick N such that the unclaimed areas are between $10^3 m^2$ and $(2*10^3 - \epsilon) m^2$. Once that happens, A and B alternate picking these areas until they run out; at that point who got the bonus region would depend on whether that number was odd or even. (If B were to make one of those regions larger than twice the minimum, it would give A extra flexibility to flip the odd/even.) Of course, B picks an even N, and subject to that condition makes the areas as large as possible (since B will end up with two more of them than A.)
So the final area is pretty close between the two; B is ahead of A by nearly $3*10^3 m^2$. (A chose to spite B by taking only the minimum size area once, making that the smallest region that B wins. There's a similar size unclaimed area.)
Incidentally,
Most of the regions of the globe will be about a millimeter wide on average, and nearly 20,000 km long. If you're looking to move there, maybe pick one of the first regions.
Edit to incorporate the OP's clarifications:
The elaborated description of "contiguous" appears to allow for enclaves (regions fully enclosing other regions, or fully enclosing unclaimed areas). In this case, the initiative (and the advantage) belongs to player A, who can create a swiss-cheese first region with unclaimed holes slightly less than twice the minimum region area. The reasoning proceeds as before but with roles reversed, except that B doesn't get to place a "big region" and A ends up with $10^5 + 3*10^3$ more land than B.