Is it possible to shorten the ordinal numbers in modern Greek?

The traditional way of doing that is to use a Greek numeral; you could use them indiscriminately for ordinals, cardinals, and in antiquity even multiplicatives. So World War II, Henry VIII: Βʹ Παγκόσμιος Πόλεμος, Ερρίκος ο Ηʹ, which are in fact read out loud as Δεύτερος Παγκόσμιος Πόλεμος, Ερρίκος ο Όγδοος, with ordinals and not cardinals. (It is “Second World War”, never “World War Two”.) This is done for names and titles.

The ordinal numbers can have a superscript inflection ending, as is done in Romance languages. That does not happen with titles, but it is optional with non-titles: you can say α[math]^{ος}[/math], β[math]^{ος}[/math], γ[math]^{ος}[/math] πρωταθλητής for 1st, 2nd, 3rd champion. Alternatively, the suffix can be hyphenated: α-ος.

These days, you will also see Arabic rather than Greek numerals, always with the inflection, and the inflection can appear with no hyphen or superscript: 1[math]^{ος}[/math], 1ος. This is newer, and if Google is any indication, that’s the most common mechanism now. In the 80s, my primary school, Sitia Second, was named Βʹ Δημοτικό Σητείας (primary schools and high schools are numbered in each town); its blog now names it 2o ΔΗΜΟΤΙΚΟ ΣΧΟΛΕΙΟ ΣΗΤΕΙΑΣ. Patras Third High School, which is old and venerable, is listed on Wikipedia as Γ’ Γυμνάσιο Πατρών; but its Facebook page names it as 3ο ΓΥΜΝΑΣΙΟ ΠΑΤΡΩΝ.