What is the word for someone who had the same job or position as someone else in the past?

Yes, if you are explicit in the subject area. So e.g. Eliot is a predecessor of Greenblatt, as a literary critic.

Predecessor primarily means what you've written: a person who held a job or office before the current holder. So, it applies when we are talking of a specific position like "(official) Critic at so-and-so organisation/establishment". For a critic in general, it is not very appropriate as there can be multiple critics existing in parallel (perhaps employed by different establishments or even in "freelance" capacity).

The terms preceded & predated are more appropriate in this context.

As a critic, Eliot preceded/pre-dated Greenblatt.

ODO:

precede VERB [WITH OBJECT]
1 Come before (something) in time.

‘She was preceded in death by her husband and is survived by two daughters.’

predate VERB [WITH OBJECT]
Exist or occur at a date earlier than (something)

‘What that shows is that there is a natural global low frequency electromagnetic signal pre-dating life on earth, so we have evolved in the presence of this signal.’

The opposite of predecessor is successor - a person who has a job, position, or title after someone else : someone who succeeds another person