I recommend purchasing tickets in advance for Jeronimos Monastery in Belem. We were there in April, and Lisbon was already getting quite crowded. We did not purchase advance tickets for the Monastery because I didn't think it would be necessary in April. We also arrived at the monastery one hour before it opened. The line to purchase tickets was short, so I was sure my strategy of getting there one hour early worked. Well, i was wrong! Our line moved very, very slowly because they allowed tour groups in as they arrived, along with people who had pre-purchased their tickets. So they only allowed a few people at a time from our line to buy tickets. We waited in line one hour before we were able to buy our tickets.
I also recommend taking the elevator to the top of the Discoveries Monument, which is across the street from the Jeronimos Monastery. There was barely a line, and you have gorgeous views of the monastery, Belem Tower, the port and the river.
We didn't go to the top of the Belem Tower because we didn't think it was necessary. If you want to do this, I recommend pre-purchasing those tickets as well.
We arrived at the Sao Jorge castle in Alfama when it opened, and waited in line for about 15 minutes.
Hope this helps.