I have a really low action and just started playing upright 2 weeks ago. Like you, I found it a bear to play the thing at first. To the point I got a 1/4 size upright (35" from the bridge to the nut). And even with the shorter scale, that made me feel like I was recovering from a stroke and having to relearn how to do basic things all over again -- things I could do without thinking on electric bass.
At the nut, the strings are about the thickness of a business card on the G, D, and A strings above the fingerboard, and the a bit less than the thickness of a credit card on the E string nut area. The strings go from about 6 mm to 11 mm from the top of the fingerboard at the bridge end of the fingerboard.
But guess what -- at two-three hours a day of playing for two weeks, it's gotten way easier. So you need to give it some time and let the muscle groups in your hands, shoulders, and arms develop. My forearm and hands were fatigued and I could feel the muscle strain (like you do when you engage in healthy exercise). At one point I felt like my hands were the size of anvils as the muscles were developing. But today I woke up with no sense that my muscles are hurting and I just played for 45 minutes with no problems at all with my hands a few minutes ago. It is getting easier.
So, I suggest you give it time. Make sure the bass is at the right height. That you are standing properly with it, and that you are using the appropriate fingering for the size of the instrument. Which normally means 1-2-4 fingering and using all available fingers to press down the strings. My teacher also told me to use an full left arm and hand "pulling" motion to press down the strings. So, its not just the muscles in your hands, it's arm muscles too. I find that helps when I start fatiguing on certain notes (like F on the low E).
Good luck. I see why many of the experienced people here kept saying there was a chance I wouldn't make it on upright because the learning curve at first is tough. Your body rejects the instrument, sadly, at first. But I am finding it is getting way easier the more I stick it out.