It is possible using only local measurements to determine one's speed relative to the Hubble flow, and hence the current cosmological time (up to an additive constant), even in a universe that's only FLRW at large scales, like ours. I know this because we've managed to do it using only experiments performed in the vicinity of Earth's orbital path over the last century or so (which is pretty small by cosmological standards) and there's nothing special about Earth's location in the universe.
These experiments involve what we call "looking at" objects that are billions of light years away "through" telescopes, but they're really local experiments because we aren't traveling out there to collect data, we're just collecting photons and neutrinos and gravitons in our immediate vicinity and inferring the existence and properties of distant objects from that.
With what we now know about the universe, the process can be streamlined a lot. You can find your velocity relative to the Hubble flow very precisely by measuring the CMBR dipole, and if you know the FLRW scale factor at all times and the initial temperature of the CMBR, you can calculate the time since the CMBR was emitted from its current temperature (after correcting for the dipole). This gives you an absolute cosmological time, albeit with its zero point at the CMBR emission rather than the big bang.
In principle you can get a more accurate answer from the neutrino and graviton backgrounds, since the universe is more transparent to them than it is to light, but the same properties that make them better in principle also make them very hard to detect.
Edit: Maybe the intent of your question is that you're in a windowless room that blocks the CMBR, and don't have precise enough equipment to detect neutrinos or gravitational waves. Can you detect the large-scale shape of the universe just from the distribution of matter in the room? In this case the answer is yes if the room is large enough that the matter in it is FLRW distributed, and no if it's smaller. If the universe is FLRW at all scales then you can detect it at all scales, but technically neither you nor the room could exist in that case since they would break the symmetry.