MS SQL Express 2016 on Amazon AWS: I Can Take Database Offline but Can't Bring It Online
You are correct: your admin account is not sysadmin, but a less powerful account. This is necessary to ensure that AWS is able to limit you from doing EVERYTHING to this instance, and escaping security boundaries. 

Setting a database online

As part of your admin account, you have access to set databases offline, but bringing a database back online requires sysadmin (which you don't have). To work around this permission restriction, AWS provides a stored procedure to bring databases online. You'll need to use the rds_set_database_online procedure in the rdsadmin database

EXEC rdsadmin.dbo.rds_set_database_online 'MyOfflineDatabase'; 


This procedure is fully documented by AWS here.

Reading the error log

Similarly, you  cannot use xp_readerrorlog on RDS SQL Server because it requires sysadmin. Instead, AWS provides THE stored procedure rds_read_error_log

EXEC rdsadmin.dbo.rds_read_error_log;


This procedure is fully documented by AWS here.