Is it possible to buy just liability insurance for a week-period to be able to drive someone else's car if that car is insured?

Coverage under the owner's liability policy is unlikely. This is because when the owner bought the policy, the insurer did not have the opportunity to assess you as a risk (such as by your driving history), and a premium was not charged or paid for that additional risk. However, you may be able to be endorsed upon the owner's policy, but you would need to go through the underwriting process and an additional premium would likely have to be paid. Alternatively, you could obtain "non-owners coverage" which affords to you liability coverage regardless of the car that you drive.

NO. If the owner of the vehicle has insurance and you have their permission to drive the vehicle then their insurance would cover you. Otherwise the owner of the vehicle could add you to their policy.