Putting on sackcloth had as its purpose—on the one hand—a practical purpose… it meant that the one wearing sackcloth, could spend all their time in prayer instead of having to spend time on clothing, hygiene and other processes designed to deal with the outward appearance, so it was useful from that perspective. However, the other function of sackcloth was to do with its outward appearance, it served as a signifier to say “look, I am sad about this.” Like wearing black at a funeral, wearing sackcloth was a way of outwardly displaying the inward darkness of sin.
We don’t see anyone in the New Testament wearing sackcloth, but what we do see is individuals and churches taking time over prayer. Think of the church in Acts 1 “all joining together constantly in prayer” or in Acts 12 when Peter was in prison, the church gathering together to earnestly pray for him. That’s the equivalent of sackcloth in terms of it being a dedicated time, where nothing else is thought about but prayer.
As for the visual benefits that came from sackcloth, I don’t know if there is an equivalent in our modern age. Worth thinking about. If you have something that you could wear (an elastic-band around your wrist?) that would remind you to pray, as well as to show the Lord that you want to focus on the need for humble-worship, then that might be worth thinking about.
Then there’s the dust on the head, which again is difficult to find an equivalent for. And yet, we go to the core meaning when it comes to dust on the head and the core meaning is humility. The dust—or the soil, or the dirt—from the floor is about as low as you can get, and so, if you’re putting that on your head, then you’re putting yourself on the same level as the dirt on the ground. And so, whereas we might not put dirt on ourselves in our own day, we can certainly seek out an attitude of humility.