Given a von Neumann Algebra $\mathcal{M} \subset \mathcal{B}(\mathcal{H})$, one defines a normal linear functional on $\mathcal{M}$ to be an ultraweakly continuous linear functional. Usually it's proven that this property is equivalent to preserving suprema of bounded increasing nets/being completely additive. However, it is important to know that $\varphi:\mathcal{M} \to \mathbb{C}$ is ultraweakly continuous on $\mathcal{M}$ iff it is only ultraweakly continuous on the unit ball of $\mathcal{M}$ (e.g. in proving that the set of normal functionals is a Banach space). I couldn't figure out a proof of this fact, the literature somehow ignores it/considers it as trivial. Any reference/clear proof will be greatly appreciated.
This of course can't be true for the usual WOT topology (on the unit ball continuity in both is the same), but in all attempts of proving this fact I didn't see a difference between using WOT and the ultraweak topology, so I couldn't devise a full proof.
Let $a_\alpha$ be a bounded positive increasing net with least upper bound $a$. Then $a_\alpha\to a$ in the ultra-strong topology, which is stronger than the ultra-weak toplogy hence $a_\alpha\to a$ ultra-weakly. It sounds like you know this statement, but if you do not you can consult Lemma 2.4.19. in the book by Bratteli and Robinson.
Note then that $a_\alpha/(2\|a\|)$ is contained entirely in the unit ball and it converges to $a/(2\|a\|)$ which is also contained in the unit ball. As such if $\varphi$ is ultra-weakly continuous on the unit ball you get that: $$\varphi(a_\alpha) = 2\|a\|\,\varphi(a_\alpha/(2\|a\|) \,)\to 2\|a\| \,\varphi(a/(2\|a\|)\,) = \varphi(a),$$ so $\varphi$ preserves suprema of bounded positive nets, hence is normal. (You did not include "positivity" in your formulation, but if this bothers you then changing the net to $\sup_\alpha\|a_\alpha\|\Bbb1 - (a-a_\alpha)$ is a way to make a bounded increasing net into a bounded increasing positive net.)