How long did fascism endure after his death?

It didn't - at least, not officially.

Mussolini was captured, and then executed on 28 April 1945: the war in Europe would've ended just a few weeks later.

Politically, he didn't survive the vote of no-confidence by part of his own Grand Council on 25 July 1943 and was sacked by the King that very same day (Marshal Pietro Badoglio was appointed PM in Mussolini's place, then Victor Emmanuel's son replaced him as "Lieutenant of the Kingdom" in early 1944). Both the Fascist militia as well as most laws enacted by the regime were repealed, albeit this had little effect on your average Italian.

True - Mussolini was later freed by the Germans only to be put in charge of the Italian Social Republic (a puppet state exercising nominal control over Central/Northern Italy) but his regime was propped up by the Nazis and it didn't last very long despite the support of many die-hard supporters.

By the time of his death, most of Italy had already been liberated by the Allies with the help of the partisans, and it was pretty much under occupation; those who associated with him either faded into obscurity - many were never tried - or simply found a new political alignment. After the war, both the Fascist Party and and its eventual reformation were outlawed, but most of its adherents simply kept quiet... a few would actually go on to found a number of far-right parties which were Fascist in all but name.

Which system carried on Italy after fascism?

Italy was a monarchy - actually, it still was one for the whole duration of the regime (with Mussolini being, at least technically, a Prime Minister) - but support for the House of Savoy had dwindled due to the war and Victor Emmanuel's accommodating stance towards Fascism. This lead the new King, Umberto II, to indict a much discussed referendum in which many alleged irregularities as well as episodes of deliberate suppression of pro-monarchic activity occurred.

Italy became a Republic on June 2, 1946, and it's been one ever since.