Hi and welcome!
I walked my first camino last year, starting in Roncesvalles in March.
There is no time in April that is too early to start, weather is so unpredictable that you just have to take pot luck. By April the sun is very nice and warm when it shines but you could get anything else. My advice would be look at how Easter slots in. It is a great time to be walking, some parts of the camino get busier because of the Spanish who walk over that period but the worst part is after Sarria and that should have got better by the time you get there as Easter is fairly early in April (8th). If you start early in April you may be able to take in some of the Holy Week processions, these are well worth seeing. I know they are particularly special after Leon, don't know about before that but if you look up some of the major towns their web pages will probably give information.
Pamplona to Santiago in 30 days maybe pushing it a bit, it will be hard work, some very long days and no time to factor in those odd happenings of the Camino. I did Pamplona to santiago in 31 days, I had a couple of short days, I also did a couple of 34 kms! Does the 30 days include your travel, if it does you would probably do better to start about Logrono or cut out a bit later on.
Buen camino
Sue