Is it possible to partially decide when will halt given any arbitrary input?
The halting problem is not decidable, but it is still partially decidable (other terms for this are "recursively enumerable" and "Turing recognizable"). So, you are right that both of these problems are equivalent to the halting problem. That is why they are undecidable, but still partially decidable.

You may want to review the definition of "partially decidable" in your textbook or notes.