HI @Alex_Batman and welcome to the Forum 
The best way to have someone to be able to access only tasks assigned to them is to manually add them to these tasks as opposed to entire projects. If you invite someone to a task level, they won’t be able to see the entire project, but just the tasks you have added them to!
You can learn more about this in this handy article: Asana tasks - understanding the basics | Product guide • Asana Product Guide
There also some similar discussion in the Forum you might find helpful, I’ve listed a few just below:
- Restrict user access
- How to limit access inside a project
- Limiting access for an assignee to the rest of a board
- Removing full granted access - #2 by Mark_Hudson
Hope this helps! Have a great Friday 