Given that many sums over the naturals (e.g., $1^2 + 2^ 2 + \ldots$) can be extended to sums over the primes using integration and the prime number theorem, it seems natural to look at sums that would be difficult to extend. The alternating sum $1 - 2 + 3 - 4 \ldots$ has an elementary sum; so consider the corresponding sum on the primes $2 - 3 + 5 - 7 + \ldots P_n$. For $n$ even, this is the negative of $(3- 2) + (7-5) + (13 - 11) + \ldots$ which is the negative of the sum of half the prime gaps between $2$ and $n$. But the sum of all the prime gaps telescopes to $P_n - 2$. Thus, heuristically a good estimate for the alternating sum appears to be $-P_n /2$ which by the prime number theorem is asymptotic to - $n \ln n/2$. Similarly, for odd $n$, a similar estimate is $-P_n/2 + P_n = n \ln n/2$. I coded up this estimate (actually, my code used the estimate due to Dusart of $P_n = n (\ln n + \ln \ln n - 1)$) and found errors less than 2% for $500 < n < 78, 401$ for both $n$ odd and $n$ even.
The question remains: is there a proof that the sum is asymptotic to $n \ln n /2$ for n odd and to $-n \ln n /2$ for $n$ even?
One of the comments refers to a limit by Ruiz http://www.primepuzzles.net/puzzles/puzz_211.htm. One of the posts there suggests that the Ruiz limit holds for prime powers as well. I have no idea whether the Ruiz result is proved (there seems to be some doubt) but the same heuristic argument above seems to extend to alternating series on prime powers $2^k - 3^k + 5^ k + ... P_n^k$. Again, for $n$ even this is the negative of the sum of the gaps $(3^k - 2^k) + (7^k - 5^k) + \ldots$ which is half of the sum of $(3^k - 2^k) + (5^k - 3^k) + (7^k - 5^k) + \ldots$ which telescopes to $P_n^k - 2^k$. This suggests heuristically again that a good estimate for the alternating sum to power $k$ is $- P_n^k/2$ for $n$ even and $P_n^k/2$ for $n$ odd. Or in terms of $n$, $-(n \ln n)^k/2$ for $n$ even and $(n \ln n)^k/2$ for $n$ odd. The OEIS has a page for $k = 1$ and some conjectures but not seemingly related to the sum. There does not seem to be an entry for $k = 2$ (the partial sums $4, -5, 20, -29, \ldots$)