Structural Coherence of Asteroids
Many asteroids are not solid blocks of rock, but rather, they are 'rubble piles'—collections of rocks and dust loosely held together by gravity. This poses a significant challenge when it comes to spinning up an asteroid to create artificial gravity. The forces involved would cause such a rubble pile asteroid to disintegrate if it spun faster than every 2.2 hours; Cruithne is 5km wide and the artificial gravity would be only 0.00016g.
In order to proceed with your plan, you'd first need to determine the structural integrity of 3753 Cruithne. If it's not a solid body, you might have to consider ways to solidify it, which could involve anything from compacting it under its own gravity to using some sort of binding agent—though these are speculative solutions and would present their own challenges.
In the vacuum of space, a spinning object should, according to Newton's first law of motion, continue spinning indefinitely unless acted upon by an external force. However, in reality, several factors could slow down the spin over time.
One of these is the YORP effect, where uneven heating of the asteroid's surface by the sun causes a torque that can change the object's rotation rate and axis orientation. Gravitational influences from other nearby celestial bodies could also affect the spin.
To maintain the desired spin rate, you would need a method of propulsion that can apply corrective forces as needed. Solar sails, electromagnetic tethers, or other propellantless propulsion methods could theoretically be used for this purpose. However, the scale of the operation and the precision required would be substantial.
Creating a habitat with Earth-like gravity is just one aspect of making an asteroid livable for humans. A host of other factors would need to be considered and engineered.
For example, the habitat would need a breathable atmosphere, which means not only having the right mix of gases (primarily nitrogen and oxygen), but also maintaining the right temperature and pressure.
Protection from radiation is another major concern. On Earth, our magnetic field and atmosphere shield us from most of the sun's harmful radiation. On an asteroid, you'd need to provide this protection artificially, perhaps by burying the habitat under several meters of rock or using some sort of magnetic shielding.
Lastly, resources such as water, food, and energy would need to be provided. While some of these could potentially be extracted from the asteroid itself or brought from Earth, sustainable methods of production would likely be necessary for a long-term habitat.