According to The Bushranger over at HP&CA there were three variants, the Tu-344A with 10 seats, the Tu-344B (shown in hesham's 3-view above) would have had 24 seats, & the Tu-344C would have had 12 seats and long-range fuel tanks.
How extensive were the modifications needed to convert the Tu-22M into the Tu-344? The fuselage seems to have a bulge in order to hold the passenger cabin. Who would have operated it? Would it have been used like the MiG-25 Business jet, to transport the Soviet Union's top brass over long distances, quickly, or would it have been used in mainline passenger service? The former seems most likely to me, as it carries to few seats to make it economical in passenger service unless the tickets were wildly expensive.
Also, would there have been any problems with the sonic boom?
"In the late 1990s, Tupolev introduced a new Tu-344 project aimed at rebuilding the Tu-22M-3 military bomber with a commercial supersonic aircraft. There was room for 10 to 12 passengers in the modified hull. According to the assumptions, the variable geometry used should have brought a higher range and greater efficiency compared to classically shaped aircraft. However, in the current circumstances, the return on investment in such a project is more than debatable."