The basic answer is: by getting records of measurements that were guaranteed to be taken at the same moment -- namely, at midday at either of the equinoxes, or either of the solstices.

It's straightforward to work out these times and dates using a gnomon (assuming you do actually take measurements), and because Syene and Meroë are on the same meridian (more or less) that guarantees that midday is the same time for both of them.

There are some bits of misinfo/misperceptions here, which are a little bit tangential to the main question, but it's probably worth addressing them anyway.

• Eratosthenes didn't take the measurements himself. People had been publishing gnomon readings taken at Syene and Meroë for decades, since Egypt came under Ptolemaic control, and this may well have been standard practice earlier too.
• It wasn't Cyrene, which is a long way to the west and definitely not on the same meridian. The premise of Eratosthenes' measurement is that you measure shadows taken at two places at the same moment, a known distance apart. With an east-west separation he had no way of getting measurements at the same moment (except at unusual moments like lunar eclipses, but then you don't have the sun giving you nice observable shadows...)
• The actual locations involved vary depending on which source you read: either Syene and Meroë (in Sudan), or Syene and Alexandria. As it happens the north-south separation is nearly the same for both pairs of cities, so the angle measured is the same (7.2 degrees).
• Both pairs of cities are a bit less than 800 km apart, and that isn't the distance Eratosthenes used as the basis for his calculation: that's Carl Sagan misreporting the distance to make Eratosthenes look good. The distance that E. actually used was 5000 stadia, which converts to around 925 km -- possibly as low as 900 or as high as 960 km depending on where you get your conversion rate, but definitely within that range.