Only get specific columns

Sure. Projection does this:

var q = from r in Context.TableName
        select new 
        {
            Id = r.Id,
            Status = r.Status
        }

Here’s an actual example (obviously, my DB has different tables than yours). I added my EF model to LINQPad and typed the following query:

from at in AddressTypes
select new
{
    Id = at.Id,
    Code = at.Code
}

LINQPad shows me that the generated SQL is:

SELECT 
    1 AS [C1], 
    [Extent1].[AddressTypeId] AS [AddressTypeId], 
    [Extent1].[Code] AS [Code]
FROM 
    [dbo].[AddressType] AS [Extent1]

None of the other fields from the table are included.

Responding to updated question

Your columns argument says it takes a type T and returns the same type. Therefore, the expression you pass must conform to this, or you need to change the type of the argument, i.e.:

public IEnumerable<U> GetBy<U>(Expression<Func<T, bool>> exp, Expression<Func<T, U>> columns)
{
    return Table.Where<T>(exp).Select<T, U>(columns);
}

Now the expression can return any type you care to use.