Why is 得 put before 我 in 搬家公司只管大件，小东西还得我亲自动手?
Is it OK and if that is the case, how can I know in what case should I put 得 before the subject?

This is OK. You can change the order of words for the purpose of emphasis; note that the object being emphasized changes according to the position of 得. e.g.

我还得亲自动手（搬）小东西。
小东西我还得亲自动手（搬）。 =&gt; for emphasizing the object, i.e. 小东西
小东西还得我亲自动手（搬）。 =&gt; not only emphasizes the object but also the subject, i.e. 我

BTW 得 is not the verb; as you said it means &quot;need&quot;, &quot;have to&quot; or &quot;must&quot;, the verb is 搬 which is omitted. This is a kind of 宾语前置 (object fronting), i.e. change from S-V-O structure to O-S-V structure.