The judge has already been in the public spotlight several times for various controversial reasons, so it doesn’t seem like all of this attention is affecting him in any negative way. In fact – and this is just a guess based on observation – he seems to take some pleasure in being able to expose these nuances of the system.
Trump has shown that he holds some strong opinions and he’s willing to go quite far in enforcing them though, so hopefully this isn’t going to lead to any negative repercussions for Judge Robart. Sure, his position gives him quite the strong shield, but it’s hard to say what kinds of cards Trump may hold in his hand at the moment.
In any case, anything Judge Robart does is sure to be examined in a lot of detail by the media, and we’re likely going to see the press paying a lot of attention to his actions in the near future.
This could also include possible attempts to smear his name, so anyone following his situation should take new developments with a grain of salt. It’s very likely that we’re going to see some controversial events sooner or later, and people should be prepared with a critical outlook on the situation. It certainly won’t help that Judge Robart has shown that he can be quite private and non-vocal at times.