Cold numbing extremities is partly due to cold actually at the extremities, and partly the blood reaching them not being very warm.
I've often seen winter runners wearing gloves and short sleeves. I've even done it myself, and occasionally cycle with full finger gloves but bare arms. If you run fast (unlike me), wind chill is an issue, as it is for me on a bike; in both cases the hands are very exposed to the wind. So a pair of thin, wind-resistant gloves might be a good idea - or slightly thicker ones if you already had something - but with full fingers.
The other side of the issue is the wrists - pulling your sleeves up is good for cooling, but exposes the blood vessels that run near the surface. You may need something as simple as pulling them back down again sooner than you thought, but if you go for gloves, ones that cover your wrists would be a good idea.
If, like me, you have long arms, that gap between sleeve and glove can let a lot of heat out. Running tops exist that have a hole to hook over your thumb - I have one that I more often use as a cycling base layer to close the gap. Warm wristbands (1980s style) might be worth a try too.
If you tend to overheat on the uphills in cold weather, a top that unzips near the throat is probably better than rolling up sleeves (or you might do both) - but remember to zip it up as soon as you cool down. That deliberately cools closer to your core.
Rewarming with your hands tucked under your arms does work, but it's not very quick to get the feeling back, and can be awkward or impossible if you need your hands. It would be better to use that at the first signs, more for prevention. So one thing you can do is check your fingers are behaving properly as you go along to know if you need to warm them; the tiny effort of wiggling them will also warm them a bit