Learn Russian: Мы как-то решим эту проблему.

It would sound unnatural here. Words with -либо mean "in any way at all" and are quite formal. They are mostly used in questions, negative sentences, or conditions. For a standard affirmative statement like "We will solve it somehow," you should use как-то or как-нибудь.

Could I use как-нибудь instead of как-то in this sentence?

Yes, but it changes the tone slightly. как-то suggests you are confident there is a specific, real way to solve it, even if you do not know it yet. как-нибудь sounds much more vague and casual, like "we will figure something out eventually" or "any old how".

How do we know решим means "will solve" rather than "are solving"?

решим is the perfective counterpart to the imperfective verb решать. Because perfective verbs focus on a completed result, their conjugated forms always refer to the future. To say "we are solving" right now, you would use the imperfective present-tense form решаем.

Why is как-то placed before the verb instead of at the end of the sentence?

In Russian, adverbs of manner usually sit just before the verb they modify. You could put it at the very end (Мы решим эту проблему как-то), but that would make "somehow" sound like an afterthought rather than a core part of your plan.