What's shown in the task grid is what prints. To print fewer rows, filter the grid first.
Filter by date
Double-click the Start or Finish column header. A filter appears where you can choose the date range to show.
Filter by resource
Double-click the Resources column header and choose the resources to show. You can also use View > Filter by Resource.
Hide completed tasks
Open the View menu and untick Show Complete Tasks.
Then print as usual. See Printing With gantterAi™.