Yes, prayer of healing would take 100 combat rounds to cast, if you tried to cast it during combat. Additionally, the cleric would also have to avoid being attacked to maintain Concentration: getting hit would initiate a Constitution Check against DC 10 or half of the damage taken, whichever is higher.

For this reason, spells that take more than an Action for their casting time aren't typically used during combat. Instead, use them during periods of narration, after combat, or in anticipation of combat. Your in-combat healing comes from faster spells like cure wounds and healing word.

Caveat: an additional in-combat healing method could be a variant human with the Healer feat, but the scope of the question is the Phandelver Starter Set.

You have to concentrate to maintain the spell, and spend your action each turn casting the spell

Longer casting times are noted in the PHB and Basic rules. I'll quote the relevant text from D&D beyond (the official 5e web toolset), since it is the easiest source to search online.

Longer Casting Times

Certain spells (including spells cast as rituals) require more time to cast: minutes or even hours. When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so. If your concentration is broken, the spell fails, but you don't expend a spell slot. If you want to try casting the spell again, you must start over.

You are still free to use your bonus action and reaction as appropriate

None of the rules for longer casting times here directly prevent you from using an available bonus action, reaction, or movement on your turn. So, you may use them as appropriate on your turn without your spell failing.

Using movement doesn't take an action, so you can still move and cast the spell

As noted in the PHB, basic rules, and D&D Beyond:

On your turn, you can move a distance up to your speed and take one action.

Moving on your turn doesn't take an action, and it doesn't break concentration. So unless the spell or some other effect requires you to stand still, you can still move on your turn.

You are limited as to casting spell with your bonus action though:

As noted in the rules on bonus action spells:

You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

Considering that you must continue to use your action each turn to cast the original spell, unless that spell is a cantrip, you wouldn't be able to cast a bonus action spell in the same turn. Luckily, spells that require your reaction don't have any such limits, so you can use those as often as you are able.