https://academic.oup.com/gh/article-abstract/8/2/229/626890?redirectedFrom=fulltext
An article about the Habsburg navy in the 19th century. Looking in Sondhaus' two books (Navies in Modern World History; and Naval Warfare 1815-1914: Warfare and History) if there's anything remotely Austrian (specifically about Friedrich)
Okay, to answer my earlier question about the use of the Austrian navy in the war with Italy, a) Naples had a fleet (she started laying down screw-propeller ships in the 1850s - before that she'd had a bunch of obsolete armoured paddle steamers (much like Russia, Prussia and Sweden did at the same time)) and b) Austria's navy had already been used in the 1820s against the Neapolitan Revolt (or rather it was supposed to be, but the army managed to put it down by themselves). Sardinia
had a navy of 8 frigates (I'm not sure if this is including the 3 60 gun warships mentioned).
A big enemy of the Austrian navy was apparently Metternich. After the Napoleonic Wars he tried to sell off a lot of ships (but found no buyers) and so many were sent to be broken up or burned. He saw the problem with this when the revolt in Naples happened in 1820, and again with the Greek Wars of Independence (since the Greek "navy" started preying on Austrian/Italian merchantmen), and this was why Austria was apparently the only European power to side with the Ottomans rather than the Greeks.
The Austrian navy was expanded in the 1840s Muhammad Ali crisis to include 2 (new) frigates, 2 (new) corvettes and several brigs (under a Rear-Admiral Bandier). The Austrian navy, under the leadership of Friedrich played a role at the battle of Sidon, the Austrian corvette
Clemenza oversaw the surrender of Tripoli, and Friedrich himself was present at the bombardment (and seizure) of Acre