Learn Spanish (Latin America): Ojalá baje el alquiler este año.

Because it is in the subjunctive mood. The word ojalá expresses a strong wish or hope, which triggers the subjunctive form of the following verb. Since bajar is an "-ar" verb, its "it" form (referring to the rent) changes its ending from -a to -e.

I learned to use Ojalá que... Why is there no que in this sentence?

After ojalá, the word que is completely optional. Both Ojalá que baje el alquiler and Ojalá baje el alquiler are perfectly correct and mean the exact same thing. Leaving out the que is very common in everyday conversational Spanish.

Why is the verb placed before the subject (baje el alquiler instead of el alquiler baje)?

In Spanish, word order is much more flexible than in English. It is very natural to put the verb before the subject, especially right after expressing a wish with ojalá. Saying Ojalá el alquiler baje is grammatically fine, but putting the action first sounds much smoother and more native.

If bajar means "to go down", how would I say "to lower" the rent?

You would use the exact same verb! Bajar can be used both when something decreases on its own (like the rent going down here) or when someone actively lowers something (like a landlord lowering the rent). You can also use it to mean "to download" a file from the internet.