I love this anime as well. I used to think he was one of the worst male protagonist EVER but I'm being drawn to him and his struggles now and I'm so close to falling in love with his complicated weak character. He constantly beats himself up and only wants the approval of his father. He's very insecure about himself and thinks of what people think of him too much. Overall his character has WAY too many problems that's why I find his pairing with Asuka so good even though she has her problems too. He's an introvert and she's an extrovert hence they balance each other evenly and it gives them a great dynamic. She pulls him out of his depression or self-doubt and he keeps her grounded from being to over confident or cocky. I never realized the detail put into the pairing now that i'm watching it over again.
One of the episodes mentions that Shinji has what's called the hedgehog's dilemma, he wants to be close to people but keeps his guard up to prevent himself from getting hurt. I find this to actually be similar to Asuka's character, too. The difference is, Shinji beats himself up and closes himself off to protect himself and Asuka goes more on the offensive.
That's what I enjoy so much about the two halves of the series. The first half, with the 'monster of the week' format, builds up Shinji's confidence and breaks down Asuka's colder exterior. Then when all hell breaks loose in the second half, it puts the characters through the ringer and that 'hedgehog's dilemma, along with other displays of mental illness, come back 10 fold.
It's one of the reasons why I enjoy Shinji's character, his journey and also why I find the finale of the movie End of Evangelion rather uplifting.