Google the Mazhandu Family Bus Company...they operate reasonably comfortable buses all over Zambia, not sure if to the North Eastern border, but you can try them. They may only go from Mpika (see map). You can book by email; take the 'business class' bus for more room.
The journey from the border (Tunduma is the Zambian border post, Mbeya in Tanzania) to Lusaka will be very, very long...I took the train years ago and that was slow. . I would allow up to 12-15 hours, depending on the time of year-the distance is over 750kms. If the Tazara train is available at Mbeya, jump on-it is slow but more comfortable.
You can buy a visa at the border(from officials) and will find no shortage of money changers-careful. Go to a forex or bank preferable.
Mazhandu run buses from Lusaka to LIvingstone 6 times a day. Get the business bus..around 80,000 KW. Takes about 6.5 hours.
Reply if you want more specifics.