Of course you don't need a cursor for this. You can use sys.columns and sys.objects to generate dynamic sql. Then simply execute it. Once you are satisfied the dynamic sql is what you want feel free to uncomment the last line.
----BE WARNED!!!!----
If you change column names your views, stored procedures, functions etc will all be broken.
declare @CurrentColumnName sysname = 'asdf'
, @NewColumnName sysname = 'qwer'
, @SQL nvarchar(MAX) = ''
select @SQL = @SQL + 'EXEC sp_rename ''' + o.name + '.' + c.name + ''', ''' + @NewColumnName + ''', ''COLUMN'';'
from sys.columns c
join sys.objects o on o.object_id = c.object_id
where c.name = @CurrentColumnName
select @SQL
--exec sp_executesql @sql
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