Since the Savoy candidacy had been floating around since 1868 and the king of Italy was very keen on it, it would have been a very good idea for Amedeo to start learning Spanish since then: if he had been fluent in the language on his arrival, it will make a very good impression. Keeping Prim alive is also a necessity. A marriage between Emanuele Filiberto, Prince of Asturias, and Infanta Eulalia would be a great idea (the Infanta is 5 years older than the prince, but it does not really matter) and might be a stepping stone toward a national reconciliation, but must be carefully prepared both with the aristocracy and the church (and the latter as I said might prove a harder nut to crush; OTOH if Amedeo and his government can find some way to reconcile the church without renouncing to the liberal policies which are his trademark, it might have a positive impact over the relations between the Savoy and the pope in Italy).
AFAIK, there are three crisis which Amedeo would have to weather at the beginning of his reign: the split in 1871 of the liberal party between constitutionalists and monarchists, which might be avoided if Prim survives; the Carlist uprising of 1872, which is almost sure to come but might be repressed at the beginning by a more stable government; the insurrection in Cuba, which had started in 1868 but peaked in 1872-73 when the Carlist insurrection made difficult for the central government to send troops to Cuba. I would say the last one might potentially offer the best opportunities to Amedeo: the grievances of the insurgents were quite real, and a policy of liberal reforms and appeasement might be successful provided that there is a stable government in Madrid. Note that I am not an expert in Spanish politics in the 1870s, much less in the situation in Cuba, and it is quite possible that I'm over-optimist on the benefits that a Prim survival might bring and on the beneficial influence that Amedeo might have. The problem with Amedeo is that he never showed a keen interest in politics before, during or after his brief kingship: he proved to be a courageous soldier in 1866, but he never followed a military career; he made a love marriage with his first wife, but notwithstanding this and the care that he devoted to his ailing wife (she died in 1876 of tuberculosis) he was well know for his "romantic escapades" (he could not keep it in his pants

).
He does not look the right material for a "reformer king", although he certainly has le phisique du role. What he might do though is to take a leaf from Louis Napoleon's playbook and go for a plebiscite: it would be a bold move, but potentially a successful one.