No. By the rules, the Radiant Sun Bolt attack is a ranged spell attack, not an unarmed strike or monk weapon. You also can't replace one of your ki-fueled bolts with an unarmed attack.

That said, I don't see a big issue with letting it be used as if it was a monk weapon if your GM wants to house-rule it that way.

I'm assuming that you're asking whether a barbarian that multiclassed into monk can use the monk subclass feature Radiant Sun Bolt (XGtE p. 35). (Or a monk that multiclassed into barbarian - same thing.)

If that is your question, then the answer is yes.

Rage (PHB p. 48) states the following about spells:

If you are able to cast spells, you can’t cast them or concentrate on them while raging.

The Sun Soul monk's class feature Radiant Sun Bolt states the following:

You gain a new attack option that you can use with the Attack action. This special attack is a ranged spell attack with a range of 30 feet

This ability explicitly uses one of the attacks you gain from using the Attack action. Therefore, you are not taking the Cast A Spell action (PHB, p. 192), and you are therefore able to use the class feature during your rage.

However, your DM could rule differently due to lore reasons.

The description of the Way of the Sun Soul monastic tradition states (emphasis mine):

Monks of the Way of the Sun Soul learn to channel their own life energy into searing bolts of light. They teach that meditation can unlock the ability to unleash the indomitable light shed by the soul of every living creature.

That sounds like it would be difficult to do while raging. Hence, it's within reason to rule this out, but since you could easily find an explanation why it is possible while raging (i.e. you trained doing it for so long, using it comes naturally), your DM should definitely tell you about his house rule before you choose the subclass or even start multiclassing into monk/barbarian at all.