Hi Rocket, I went to Barcelona in June of last year. I also get tired easily but we managed to get around using the local buses which are quite reasonable. There is also a tourist bus that goes to all the main places which is hop on hop off. Bit more expensive but probably worth it. If you decide to get a wheelchair getting round the streets isn’t so difficult as they have smooth pavements. depending on which month you are going, take the local bus to the beach, great to people watch with some lovely bars.
Not sure which airport you are going from but as Jan says do use the assistance both ends. We usually fly out of Gatwick and the assistance is excellent. Just book ahead with the airline. Also if you phone the airline ahead of travel, you can book your seats with no charge. Assistance is also provided at Barcelona although can be slightly more hit and miss. If you google Barcelona assistance that will give you all the information and how to access it. The bus from the airport is very reasonable and the people taking the money let me jump the queue which was a bonus. Something else to be aware of if you are flying Easyjet, the departure building when you are coming home is a long walk from where the bus drops off.
Barcelona is a beautiful city but I have to admit it was very hard on me, so much so that it was the deciding factor to get a wheelchair for future holidays.
Have a brilliant time.