On Netflix, in the first episode, at 00:43 he says it. Netflix translates it as "what a drag". It sounds to me like 「だりー」 not 「だるい」. But maybe I'm mishearing?
The dictionary says だるい means sluggish; languid; listless; heavy. Google translates it as "tired". Which seems like a good fit.
I did find this image of だるい but I'm not sure if it's the catch phrase or just another related phrase.
What is Gudetama saying? What's his catch phrase?