To move a satellite in geostationary orbit, 166°55′E, to the antipode, 13° 4' 3.2" W, what delta-V would be needed for this to be accomplished?
As for time constraints, I do not know what are the possibilities with contemporary technology. What would be the fastest? What would be the delta-V if the time required was four times slower than the fastest option already discussed?
Theoretically, you can go anywhere in GEO for an arbitrarily small ∆v - you raise your apogee a little bit, which slows you down, wait until you've phased to your destination latitude, then re-circularize back into GEO.
In practice, though, as @uhoh mentions in comments, there are stable longitudes in GEO that require more than an infinitesimal maneuver to escape from. The maximum instability according to that paper requires only about 2 m/s per year to correct, however, so I would guess that any maneuver of more than a few m/s can escape the stable nodes.
So, the decisive figure is how long you want to take to move your satellite.
If you want to go to the opposite side of Earth in a month, you need to raise your orbit to the height where you're going 29.5/30.0 times as fast as GEO, so you lose half an orbit after 30 days.
The semi-major axis of an orbit with period $t$ is:
$$a = \sqrt[3]{\frac{\mu t^2}{4\pi^2}}$$
Where $\mu$ is Earth's standard gravitational parameter. For this orbit the SMA is about 42639 km (radius, not altitude). Holding perigee fixed, you get a 35736 km by 36750 km orbit. That apogee raise maneuver is equivalent to the first impulse of an ideal Hohmann transfer, the cost of which is given by:
$$\Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1+r_2}} - 1 \right)$$
Which works out to about 17.3 m/s to raise the orbit, and the same to recircularize a month later, for a total of about 35 m/s.
To do it in a week, you need to go to an orbit that's 6.5/7.0 as fast - an apogee of 40071 km. The cost here is about proportional to the speedup - 73.5 m/s to get into or out of the phasing orbit, for a total of 147 m/s.
If you can wait 6 months, you phase a degree per day and the cost drops to about 6.2 m/s.
The really fast way would be to drop perigee to 4595km, which is an orbit that takes only 12 hours to complete, re-circularize after one orbit when you're back at geosynchronous altitude, outrunning those slowpokes in GEO -- this takes 1099 m/s on each end for a total of 2198 m/s.
