What is the best way to analyze non-quantitative mass spec hits from an immunoprecipitation pull down?
If they are available, talk to the people who analysed your data. Using their knowledge is intelligent.

I will assume two critical steps are completed. First that your results have been searched correctly. I will also assume that the peptide and protein identifications have been filtered to provide a false discovery rate of 1:100 at each level.

You can remove all the proteins that were identified in your affinity purification control. Another option is to use the Contaminant Repository for Affinity Purification (the CRAPome) which provides a formal framework for removing contaminant proteins. It is a nice resource if your data are compatible.

The remaining proteins are what you have to work with, this is the 


  list of potential proteins that [your protein] interacts with.


Prioritise these proteins based on the biology of your experimental system. If the list is small work through the function of each protein and assess how it fits with the biology you are researching.

To assess the potential functional links between the proteins you can create an in silico network. Markov clustering can highlight the presence of functionally distinct groups of proteins within a network. String, linked above, can do this. Centrality measures can indicate the presence of proteins with the potential to influence the behaviour of the network.

For larger lists you can also assess the function of the proteins in the list by performing a functional annotation enrichment analysis. There are many other things you could do but the primary goal is to keep biology at the forefront of your efforts. 

Finally you mention that repeats are necessary. Analysing the data is time consuming. If this was a test run verify that the test was successful and then run the replicates. Time spent analysing incomplete data can be spent completing the data set or ensuring the right data is collected. Quantitative data can be very useful and with care shotgun proteomics approaches can provide quantitative data but the people who run the machine need to be consulted when quantitative data is required.