I'd recommend finding a concrete reason for what the implications of waking up in the game are. I'm not familiar with Log Horizon, but I know with Overlord, there was a degree of "I'm suddenly immortal, I can do whatever I want" and "I'm going to exploit my power and scout the world for any other people online". So bring stuck in the game served to start the plot and inform the protagonists decisions. In a multi player game, there are a few reactions people are likely to have (after the mass panic and confusion subsides).
1. Try to find a way to get out. Groups will form and people will search for any possible loophole that can get them out. Characters with family to return to, jobs to get back to, etc. are likely going to be working together en masse
2. Make the best of a bad situation. People who don't believe there is a way out, or those who aren't happy with their real life, are likely going to focus their attention on making a life for themselves in this virtual world. They might even be happy to have an excuse to escape completely (though they may eventually become disillusioned and decide that the fictional world is not better than the real one, or perhaps they'll end up happy that they could reinvent themselves)
One thing you could do is let players be on either side and do whatever they want and play around. You could also focus the plotline on a single side and explore how they react and work to accomplish their goal (this has the benefit of keeping players together and in the same area). Another interesting thing you could do is focus the plot on a conflict between the people who put their effort into trying to get out and the people who think that things are good as they are. I'd imagine that neither group would be happy with the other side's attitude.
Or you could decide to not use any of that ;P If you have something different in mind for what you'd want to focus the rp on, it'd help me to understand better what you're going for if you explain what the rp will focus on. Background is helpful but you also need a roadmap for where you want to direct the plot
1. Try to find a way to get out. Groups will form and people will search for any possible loophole that can get them out. Characters with family to return to, jobs to get back to, etc. are likely going to be working together en masse
2. Make the best of a bad situation. People who don't believe there is a way out, or those who aren't happy with their real life, are likely going to focus their attention on making a life for themselves in this virtual world. They might even be happy to have an excuse to escape completely (though they may eventually become disillusioned and decide that the fictional world is not better than the real one, or perhaps they'll end up happy that they could reinvent themselves)
One thing you could do is let players be on either side and do whatever they want and play around. You could also focus the plotline on a single side and explore how they react and work to accomplish their goal (this has the benefit of keeping players together and in the same area). Another interesting thing you could do is focus the plot on a conflict between the people who put their effort into trying to get out and the people who think that things are good as they are. I'd imagine that neither group would be happy with the other side's attitude.
Or you could decide to not use any of that ;P If you have something different in mind for what you'd want to focus the rp on, it'd help me to understand better what you're going for if you explain what the rp will focus on. Background is helpful but you also need a roadmap for where you want to direct the plot