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Do you mean for putative astronomers on that object?
Indeed, There would a "main" star that slowly, over several generations, got lower or higher in the sky. There would be a night time set of stars and a daytime as well leading to a very directional and non isotropic cosmology. This could lead to some very interesting myth creation stories.
Do we really have much of an idea what happens in the outer reaches of our/any solar system?
Very little, there are no self-luminous objects out there even in the infra red. We can only really spot them by reflected light from the sun and that leads to a relative brightness of (1/10000)^2.
There are some interesting pictures of exo-solar systems via interferometry showing the formation of planets close to the host star.
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Backstop
Do you mean for putative astronomers on that object?
Yes, obviously it is probably held to our sun in a rather tenuous way--doesn't it just mean that it's more subject to our sun's gravity than to that of anything else (including a hypothesised large planet)?
Do we really have much of an idea what happens in the outer reaches of our/any solar system?