So your main issue is really, how would you scale to 1,000 users all at once.
Given the vagueness of this question and lack of knowledge about your personal app you are inquiring about, this will be pretty high level.
There are many ways to solve your problem but here is probably the most common.
Sessions are by user (as Sinisa mentioned) so they should be handled accordingly. You can save session info to a DB or via cookie (in mvc its likely via cookie).
As for handling the users you'd want to look into load balancers. There is a lot of writing and how to's on this topic but in short, load balancers allow you to have more than one server accessible via a single URL with the capability to handle extreme loads.
Database are typically a bottleneck. Depending on what your app does you'll likely want to look into caching options (memcache, redis, amazon elasticache....many options). This creates its own issues (invalidating/refreshing the cache) but it frees up some of your DB operations from being the bottleneck in your application.
Redis: What and Why? – codeburst
Amazon ElastiCache- In-memory data store and cache
A google search about caching in mvc/.net will provide plenty of links to read through.
There are additional options here including DB sharding for load balancing but now we are getting into advanced topics beyond the scope of a quick answer forum.
I'd also suggest doing searches on advanced scaling topics/scalability topics to read up on as well.