Can monetary union work without fiscal union?
Yes, and that facilitates a comparison of economies & policies.
Note that the governmental participation you outline (tending to government's prominence in an economy) is only one approach of policy making. Another, opposite approach is neoliberalism, whereby a government's role in an economy is mostly supervisory and promotes market freedom.
It is the absence of fiscal union what allows a government to make the domestic economy more competitive by lowering taxes. And in a context of monetary union, the spectrum of fiscal differences becomes more evident, which streamlines an assessment of which economies or policies are more effective. In short, the combination of monetary union and lack of fiscal union is the "best of both worlds".
for example by undertaking infrastructure projects. Does monetary union somehow limit the ability to use this lever too?
It certainly does because monetary union removes a degree of freedom: The members of a monetary union are subject to an additional constraint necessary to preserve stability of the currency they have in common.
But more important is that the constraint that ensues from a monetary union makes it more difficult for a government to go too far on policies that ultimately are detrimental or unsustainable.