Arch rivals? Well when I started supporting Juve in the '81-'82 season, Roma was clearly our main rival. Thats when Juve and Roma would take turns every year winning the scudetto. But those were great teams. The Juve team of 82 had something like 6 or 7 starters in the national team...Zoff, Cabrini, Scirea, Gentile, Rossi, Tardelli. Then Juve added Platini and Boniek to that lineup and started our great dominance of the early to mid 80s. So in essence, 1/2 our squad was world cup champions. The Roma team boasted the likes of Bruno Conti, Falcao, Cerezo, Tancredi, Pruzzo(overrated), and even a young Ancelotti. In Italy, in my mind, Roma still remains our arch rival.
In Europe, back then we played Liverpool for the european cup final. It was a disastrous event but I remember that Liverpool had beat Roma in the Cup final the year before on penalties. Bruno Conti shot over the cross bar and Liverpool became champions. We met them the following year and beat them when a l'pool defender fouled Boniek in the area, and Platini made the penalty. It was a great liverpool team, and they were probably the best european team in the 80s. Great players in every department ( maybe except for goalie, grobelaar was funny though). John Barnes and Peter Beardsley were their main weapons along with Ian Rush whom Juve purchased later but flopped in Italy(surpise).
But yeah, to sum it up, definitely Roma and Liverpool were our arch rivals in the early to mid 80s, then Juve fell off after Platini retired giving way to the great Milan side of the late 80s-early 90s.