I would say almost certainly not. I can't see any evidence the Japanese government has passed any law or made any promises to assume liability, and they are not liable under international law - the main law that governs this is the amended IAEA Vienna Convention which most but not all countries using nuclear power are signatories to. Certainly it would appear to be a pretty stupid move to assume responsibility here, and with no source for the claim it doesn't seem credible.
- Operators of nuclear power plants are liable for any damage caused by them, regardless of fault. They therefore normally take out insurance for third-party liability, and in most countries they are required to do so.
- The potential cross boundary consequences of a nuclear accident require an international nuclear liability regime, so national laws are supplemented by a number of international conventions.
- Liability is limited by both international conventions and by national legislation, so that beyond the limit (normally covered by insurance) the state can accept responsibility as insurer of last resort, as in all other aspects of industrial society.
Exclusive liability of the operator means that in the case of an accident, all claims are to be brought against the nuclear operator. This legal channeling is regardless of the accident's cause. By inference suppliers or builders of the plant are protected from public litigation in the case of an accident. Again this simplifies the process because claimants do not have to figure out who is responsible – under law it will be the nuclear operator.
Liability for Nuclear Damage, World Nuclear Association, April 2014
World-wide, almost all legal regimes transfer third party liability for nuclear accidents exclusively towards the operator of a nuclear plant. This is called “channelling” and implies that the operator of a nuclear installation is exclusively liable for damages, either legally or economically. Irrespective of their possible contribution, none of the other players – suppliers of nuclear material or fuel, transporters of nuclear material or fuel to and from the nuclear power plant, subcontractors, test operators, consultants, nuclear plant designers and constructors – bears any responsibility towards third parties in the event of a nuclear accident.
Evelyne Ameye, 'Channelling of Nuclear Third Party Liability towards the Operator: Is it Sustainable in a Developing Nuclear World or is there a Need for Liability of Nuclear Architects and Engineers?' (2010) 19 European Energy and Environmental Law Review, Issue 1, pp. 33–58
TL;DR Japanese citizens are not liable for accidents abroad under international law. No evidence of assumed responsibility.