The National Academy of Sciences undertook a study, funded by NASA, regarding near-Earth object detection and mitigation. This study looked at four mitigation techniques: nuclear, kinetic impactors, gravity tractors, and civil defense. The study did not address paint or solar sails due to the extremely low level of technology readiness of those techniques.
The results of this study were published in 2010 at https://www.nap.edu/catalog/12842/defending-planet-earth-near-earth-object-surveys-and-hazard-mitigation . The graph below portrays key aspects of their findings.
Source: Chapter 5 of the previously cited study.
The nuclear option comes in two forms: Disruption and standoff explosions. The intent of the disruptive explosions is to cause the asteroid to break into tiny parts, almost all of which will be placed on a trajectory that avoids collision with the Earth. Unfortunately, this approach will most likely leave at least one large mass that remains on a collision course.
The intent of the standoff approach is to have the weapon explode some distance from the asteroid, thereby avoiding disruption. The gamma rays and energetic neutrons produced by the explosions will cause surface material on the asteroid to vaporize, generating an impulsive change in the velocity of the asteroid body, which remains intact. The disadvantage of the standoff compared to disruptive explosions is that a standoff explosion delivers less than half of the weapon's energy to the asteroid while the disruptive approach delivers almost all of the weapon's energy to the asteroid. The advantage of the standoff approach is that it isn't disruptive. A disruptive explosion is the last resort.
Kinetic impactors are non-nuclear devices that collide with the threatening object at a very high relative velocity, thereby imparting an impulsive change in the velocity of the threatening object. These are orders of magnitude less effective but orders of magnitude more acceptable than the nuclear options.
Gravity tractors are devices that maintain a somewhat largish mass close to the threatening object by means of a low-level thrust. Gravitation makes the threatening object accelerate toward the tractor. The low-level thrust keeps the tractor and its massive payload from falling into the asteroid. This is a very slow but steady approach. Multiple decades of operational time are needed to be effective. In addition to the long time span, another downside of this and related approaches is that these approaches require rendezvous, significantly increasing the delta V requirements of such missions compared to the nuclear options or kinetic impactors.
Note that in the above graph, the level of shading indicates the effectiveness of the technique. For an object 10 km across, the only effective technique is the nuclear option, and then only if there is a decade or more advance warning. The question asks about using paint or solar sails for an object that is 10 km across, with only one year of advance warning. Nothing is effective in this extreme case.
What about using paint or solar sails? These techniques (along with a myriad of others) were dismissed out of hand by the National Academy of Sciences study due to impracticality, lack of knowledge, or lack of technology readiness. The Yarkovsky effect, which is what using paint relies upon, is something that suffers from lack of knowledge. Scientists don't quite know the magnitude of the Yarkovsky effect on asteroids. Studying this is one of the key aspects of the ongoing OSIRIS-REx mission. The extremely low technology readiness of this approach also contributes to making this a technique that, at least currently, can be dismissed out of hand from an engineering perspective.
Solar sails might be effective as an alternative to using thrusters in the gravity tractor. An extremely large solar sail to accomplish this, making this a TRL 1 (idea on paper) kind of approach. Tethering the solar sail to the threatening object falls into the sub TRL 1 territory.
