It would be very difficulty to see individual proteins and their interactions when the number is 1000. In such cases the best you can do is play with different layouts in Cytoscape to get a meaningful global representation. I have made networks with 15000+ proteins in Cytoscape, but those are not meant for looking into individual proteins and their interactions.
Here is an example of all
c.1. folds in SCOP and their relationships, which is comparable in number to what you have.
As you were already told above, adjust the stringency of your criteria to get most reliable interactions for your protein(s) of interest, and that should reduce the number of proteins and be easier to visualize. Either way you will have to play with layouts, or else any network will end up looking like a blob in your image.