It should only take several hours for an electrician to disconnect the wires and rotate the box if that was all that was involved. However, it's hard to tell what all is happening in that box. For example, it looks like there are wires coming out of the bottom of the box that are going somewhere else, and there may also be some splices in the box that are not to code. How much these issues add to the time I can't say. Also, you are aware, aren't you, that the electrician will have to cut out sheetrock above and below the box to pull the wires through and then reinstall them once the box is turned? And you are getting a permit, right?

Have to disagree with GreenDesigns who states, "they offer almost zero improvement in efficiency of a new high quality tanked electric heater. ...electrical tankless is only truly practical if designed for a new build from the beginning. Retrofitting is extremely expensive and won't give you a payback for the cost differences inside of your lifetime." I put a whole-house electric tankless in my 1906 house and will never store water in a tank again. Absolutely love it! It's mounted on the wall, directly to the side and just above the dryer. I did not bother to hide the water lines inside the wall, but since they're [mostly] hidden by the dryer, who cares? Sometimes, putting plumbing pipes in the wall is highly overrated. Mine is in the bath/laundry room and I've never found it to be unsightly. Yes, the tankless requires quite a bit of power: Mine requires two, double-pole 60amp breakers with 6g wire. So, here's where distance from the electric panel is a cost consideration.

The bulb in this socket draws enough current that the spurious voltages will not occur. I assume that there are two black/white pairs which come from the breaker box. It's easy to see which black is which. First, connect the white on this socket to the twisted whites and touch the black on the test light to each of the blacks and see if any of the blacks are hot. Once you have confirmed that at least one white/black pair is powered, turn off that breaker and make these temporary connections more secure (not just touching live wires). Turn the breaker back on and then turn on and off the likely wall switches. If none of the switches make the test light go off, then you have found a source of power. Repeat the above for all the black wires and see if any of them are switched. Finally, and I'm not sure about the following test, but if one of the black wires never shows power, it must go to another light fixture or receptacle. Connect your test light between the black power source and this unknown black wire. If your test light glows, then that black wire goes to something which draws current (perhaps a ceiling light with an incandescent bulb in it), and you can figure out what that is by observation. If it doesn't glow, then take another test light, put a regular plug on it, and plug it into each of the likely receptacles. If this second test light glows, then the two lights are in series and you know where this unknown wire goes and you can nut it to the power source wire (or to the switched wire, if you want that receptacle switched). One thing I don't know is how to trace out the white wires. If each of them act as a reliable neutral, then maybe it doesn't matter. If it does matter, I don't know how to identify which is which. Sorry for this long response; I hope I'm clear.

Your utility company will likely be the one to determine if you need a new service line to the meter (probably yes) and how it can be run. Your Service Entry cable from the meter to the breaker panel will need to be replaced (also probably yes). Discuss these with your electrician.