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

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.’