Is it known wether the following limit tends to Infinity or not? Is there any possibility for it to converge to a constant?
$$\lim_{n \to \infty} \sum^n_{p\le n} \frac{1}{\sqrt{p}} - li(\sqrt{n})$$
(Being $p$ a prime number and $li(k)$ the logarithmic integral)
I have not found anything about it. It seems that most efforts are made trying to upper-bound the asymptotic behaviour of the limit. For example, here there is a proof of this behaviour based on the Prime Number Theorem. Also, the Riemann's Hypothesis provides a tighter upper bound. But what about a lower bound?
In your linked question it is using the prime number theorem, that is equivalent to $$\sum_{p < x} p^{-1/2} - li(\sqrt{x}) = o\left(\frac{x^{1/2}}{(\ln x)^2}\right)$$ That's all we can do, the PNT has been proven in 1900 and no major step has been done until today.
It is already quite hard to show that the Riemann hypothesis is equivalent to $$\forall \epsilon > 0, \qquad\sum_{p < x} p^{-1/2} - li(\sqrt{x}) = o\left(x^{\epsilon}\right)$$
Assuming the RH the experts can do better, obtaining $O(\ln^2 x)$ as I mentioned in your other question. If you want to know the best error term, make some research you'll find easily many papers about that.