Sadly not.
If our universe was steady state, or at least expanding much more slowly, then in principle you would be able to see all the way around it, back to where you were. This also assumes that this slowly evolving, or steady state universe is also a four dimensional enclosed "bubble", like our rapidly evolving universe appears to be. If this were the case, it’s possible you could benefit from a sort of "hall of mirrors" effect and be able to measure earlier versions of your own neighbourhood.
Interestingly enough, at least in its simplest form, this only works if you keep moving around. If instead you remained completely static, then you actually fill your own sky, and therefore obscure your own view of earlier yous.
If you "unsimplify this model a bit" to propose a universe like ours but static and a bit smaller, you could receive images that had circumnavigated the universe. However, the whole principle of these enclosed hyper spheres is that space is not flat. It is therefore likely (depending on the distribution and density of material in your universe) that the accumulation of all the gravitational distortions an image encountered during its trip would probably scatter the images in ways that you could not possibly resolve. Unless, of course, you already happen to have a perfect map of all the mass distribution in your steady state mini universe to start with.
Back in the real universe what we find as we look back with more powerful instruments is that our view of history is obscured by a 380,000 year old "ball of fire", or plasma to be precise. When I say its 380,000 years old, what I mean is that it existed for 380,000 years, before it became visible, so when we look at it, the object we see was 380,000 years old at that time. It is, in fact, our own universe, on its 380,000th birthday, still very young for a universe of this type.
What we find is, for reasons that we still don't fully understand, a universe that is considerably greater in girth than age. How it has managed to put on so much bulk, and even where exactly it keeps it all, is still something of a mystery, but the result is that in whatever direction you look, you can never see all the way around. You can see as far back as the point in our history where the fire went out, and the universe became transparent, but not beyond.
Prior to the year 380,000 the universe was so hot an dense that even hydrogen atoms could not form without being instantly smashed apart again by the surrounding particles. The result is a piping hot soup of elementary charged particles that would be constantly emitting and absorbing photons like it was going out of fashion. Which eventually it did. On, it’s 380,000th birthday, or roughly thereabouts, the universe had finally relaxed enough to allow its electrons and protons to get together long enough, and with enough privacy, to go and make some hydrogen. Once they started, there was no stopping them and soon pretty much all the matter in the universe had been converted to hydrogen. This resulted in photon decoupling.
All forms of matter and radiation had been pretty intimate up until that point, but once the electrons and the protons got together, the photon were left to go their own way. Now, free from interference, they could cross the vast distances, and eventually, vast times, that span our universe. Many of the photons emitted during the decoupling phase of our universe are still travelling even now. Over the past 14 billion years of their journey, the universe itself has stretched out beneath them. Once hot young energetic blue photons are now stretched out by an ever expanding cosmos to larger, and somewhat less energetic, wavelengths. Still rippling along since almost, but not quite, the start of time.
We experience these echoes of the past as the microwave background, and now we have been able to map it with the Wilkinson Microwave Anisotropy Probe (WMAP). This image is basically a cross-section of the universe at the point where the fire went out any everything collapsed into neutral hydrogen.
Please note that for the purpose of this article "one year" simply means about 31.5 million seconds, and all dates are approximate.
