Can't query and put data inside a cursor when using variable inside the query

Cursors should be right at the bottom of your bag of techniques, used sparingly and with great care, only when necessary. I can't tell if it's necessary in your case, there's not enough code to know. But I wanted to get that out before continuing.

As a point of purely academic interest, yes, there are some ways you can do this. Two main ways:

1. Declare a cursor in the dynamic SQL, as Dale suggested. You can still use the cursor in static code which follows the declaration if the cursor is global.
2. Use dynamic SQL to drop the results into something with scope outside of the dynamic sql, like a temp table. The cursor over the temp table.

1 is just bad. It is likely to result in code which is extremely difficult to understand in future. I include it for curiosity only. 2 is reasonable.

Examples:

-- some dummy schema and data to work with
create table t(i int); 
insert t values(1), (2);

-- option 1: declare a cursor dynamically, use it statically (don't do this)

declare @i int;
exec sp_executesql N'declare c cursor global for select i from t';
open c;
fetch next from c into @i;
while (@@fetch_status = 0) 
begin
    print @i;
    fetch next from c into @i;
end
close c;
deallocate c;

-- option 2: dynamically dump data to a table, eg a temp table

create table #u(i int);
exec sp_executesql N'insert #u (i) select i from t';
declare c cursor local for select i from #u;
declare @i int;
open c;
fetch next from c into @i;
while (@@fetch_status = 0) 
begin
    print @i;
    fetch next from c into @i;
end
close c;
deallocate c;