Do you mean one long climb of 20 miles that is 4,500 feet in elevation gain (case 1)? If so, and your fitness is "a shame" you will be hurting. If you mean 20 miles of up and downs with 4,500 feet in gain (case 2) you will be hurting even more but you do get to recover a bit after each climb.

Case 1: You may be able to spin 34/23 fast enough to maintain some level of comfort (80rpm ~10mph) but even then it will take you about 2 hours up constant 4-5% grades. Vail pass (from Vail) is about 11 miles and 2,200 feet. I rode this with a compact and 11-28. I found it to be a long and somewhat hard climb (though I had already ridden up the other side). Your ride is basically double this. I don't think I would be able to do that with just 34/23 - at least not without a few breaks and probably sore knees.

Case 2: I have a ride I do with a bit over 2,000 feet over 20 miles. It's 4 short, steep climbs one way and then turn around and do them backwards. I have an 11-23 on one bike and can do this ride but I much prefer the 11-28. I'm 200lbs but I'm also in fairly good shape. I would not have been able to do this ride (and wasn't) when I first got a road bike even though I was in fairly good shape then too (though less than now). Add another 2,500 feet in those 20 miles and I think you will be hurting regardless of a 23 or 28 in back.