You cannot add a column with a default value in Hive. You have the right syntax for adding the column ALTER TABLE test1 ADD COLUMNS (access_count1 int);, you just need to get rid of default sum(max_count). No changes to that files backing your table will happen as a result of adding the column. Hive handles the "missing" data by interpreting NULL as the value for every cell in that column.
So now your have the problem of needing to populate the column. Unfortunately in Hive you essentially need to rewrite the whole table, this time with the column populated. It may be easier to rerun your original query with the new column. Or you could add the column to the table you have now, then select all of its columns plus value for the new column.
You also have the option to always COALESCE the column to your desired default and leave it NULL for now. This option fails when you want NULL to have a meaning distinct from your desired default. It also requires you to depend on always remembering to COALESCE.
If you are very confident in your abilities to deal with the files backing Hive, you could also directly alter them to add your default. In general I would recommend against this because most of the time it will be slower and more dangerous. There might be some case where it makes sense though, so I've included this option for completeness.