Firstly, I would argue that Asians handle modernity in much the same way as the rest of the world.
Some Asian countries, such as Japan, South Korea, Singapore, Indonesia, and India have huge tech sectors and a booming modern economy, as well as an advanced and arguably better-balanced society than most countries on this planet.
But assuming they do face challenges with regards to modernity, the answer and solution would have to be specific to the relevant challenge you have in mind.
For example, let's say the challenges that an Asian country faced were divided into the following categories:
- Social
- Political
- Economics
- Ethical
It could be suggested that any challenge within the above fields could be tackled through a combination of the following:
- Identifying and relying on their own historical and ethical points of reference
- Adopting a model that is influenced by the ethical values and spiritual beliefs the nation holds dear
- Learning from the success and failures of their own nation, and those of other countries around the globe.