All the above suggestions are good. Google will easily turn up sources for the cooling towel (you just dampen it, swirl it around in the air, and it gets quite cool) and bead-filled ties that go around your neck and cool that area by evaporation (CoolTies and competitors). I have used the latter and tested the former.
I'm sure many people wouldn't like the CoolTie-style products because they mean wearing a sausage-shaped thing around your neck, but in my experience they remain damp all day and do help you stay cool.
The cooling towel is nice because you can use it to wipe your face. However, once you get it wet, you need to have a way to carry it around with you.
You cannot count on air conditioning in all indoor spaces (it's not always available in shops, cafes and museums), so it is very, very valuable to have lodgings (which must be air conditioned) conveniently located near the predominant sightseeing area. That will make it easier to return to your room to cool off. Be cautious about planning day trips, because you will not have access to your room during them. A tip: on really hot days, a door propped open is usually an indication that there's no a/c inside; if you need to cool off, find another shop/cafe/etc.
I want to reinforce the drink-water advice. Not being a fan of ancient ruins, I haven't had to deal with long visits to exposed, non-urban areas in the summer, but I developed heat exhaustion as I walked home on a 90F day, after donating blood. It came on very, very suddenly, and I was lucky to make it to my air conditioned apartment. (It was during the pandemic; nothing was open and there were none of the usual street vendors selling cold drinks.) I was moving at about 1/3 of my normal speed; it was scary.