Japanese (and other asians) wearing surgical masks is a common practise that predates the Fukushima accident.
Regarding radioactive radiation:
No, a thin piece of paper will not protect you, except maybe Alpha radiation:
Protecting yourself from external exposure to alpha radiation is easy, since alpha particles are unable to penetrate the outer dead layers of skin or clothing.
However, tissue that is not protected by the outer layer of dead cells, such as eyes or open wounds, must be carefully protected.
- Greater concern about beta particles. They can burn the skin in some cases, or damage eyes.
- Greatest concern is about gamma radiation. Different radionuclides emit gamma rays of different strength, but gamma rays can travel long
distances and penetrate entirely through the body.
Gamma rays can be slowed by dense material (shielding), such as lead, and can be stopped if the material is thick enough. Examples of shielding are containers; protective clothing, such as a lead apron; and soil covering buried radioactive materials.
Wearing a surgical mask may however help against inhaling radioactive dust. It depends on how fine the dust is. But some protection is still better than no protection.
Regarding viral infection:
WHO - Advice on the use of masks in the community setting in
Influenza A (H1N1) outbreaks (2009)
In health-care settings, studies evaluating measures to reduce the
spread of respiratory viruses suggest that the use of masks could
reduce the transmission of influenza.
the community, however, the benefits of wearing masks has not been
established, especially in open areas, as opposed to enclosed spaces
while in close contact with a person with influenza-like symptoms.
Nonetheless ... using a mask can enable an individual
with influenza-like symptoms to cover their mouth and nose to help
contain respiratory droplets, a measure that is part of cough
Using a mask incorrectly however, may actually increase the
risk of transmission, rather than reduce it.