There are two bus services between Salerno and Matera; neither one takes 4.5 hours.
The Miccolis bus takes 3 hours 10 minutes. You can see a photo of the coach on their website. It looks very comfortable.
https://booking.busmiccolis.it/
Trenitalia runs a FrecciaLink bus that can only be booked together with a Frecce train ticket to Salerno. A ticket from Rome, Caserta, or Naples will work. The bus part takes 2h 55 minutes.
There is a photo of the bus on this webpage, but it is from 2016 and the bus may be different now.
https://www.fsitaliane.it/content/fsitaliane/en/media/press-releases/2016/6/28/s-born-the-Frecciarossa-reaches-Matera-and-Potenza.html
You can book this one on Trenitalia, using Napoli Centrale ( or other appropriate station) as the starting point and Matera as the end point.
Cost for either service is under 30€.
We visited Matera for a few hours in early March (part of a Road Scholar tour) and hope to return next year to spend a couple of nights and explore more on our own.