Is it viable to earn in-game ISK to buy premium

I'm going to say that your question is subjective and depends on the kind of player that you are, as Ellesedil mentioned. It might be fun for one player to grind 20-30 hours per week for their monthly PLEX, and another player would absolutely despise it.

On that note I'll say that it is entirely possible for a new player to earn enough ISK in-game on a regular basis to afford the omega (premium) account. There are several ways of doing this, exploration and combat sites likely being the easiest to get started. As you become a more experienced player, it will become easier to make the ISK each month to pay the subscription.

Will you (the average new player) enjoy this? Will it be a rewarding and satisfying use of your time? Absolutely not. Think about it like working a second job, if you're spending 20 hours per week grinding for 1.8 billion that you exchange for the equivalent of $15 in real life currency -- you're cheating yourself. It would be a far more efficient use of your time to spend 1 hour in real life earning the $15 that might have taken you 40-60+ hours in-game. However, if the sense of reward you get from achieving the goal each month is a reward in itself, then the game has achieved its intent of being what it is, a game that's fun. What I would instead is take that 1.8 billion and use it to buy ships, skill injectors, etc. In my experience this makes the game far more enjoyable and opens up new opportunities.

You don't need the premium access to have fun in the game. Most of the content will be available to you as an alpha character. After you've played for around a year however, you'll definitely want to upgrade your account in order to join in activities that require more specialized ships and fits. Some more experienced groups will require you to be an omega character. If you're already spending countless hours each month playing the game, why not spend $15 to make your experience that much more enjoyable?

After 10 years of playing EVE, my advice for making ISK as a new player would be to 1. Do wormhole faction relic site exploration in c1, c2, c3 wormholes. You can easily make 200 million on a lucky day with a few hours grind. If you want more action (or less depending on the perspective), I would recommend joining a nullsec alliance and starting in combat sites in a ship such as a navy vexor, or whatever meta fit people are using. You can also run highsec exploration combat sites, but it's more competitive. Regardless of which option you choose to make ISK, your ability to generate income scales with how engaged you are in learning from other players (in my experience). Get involved with a player alliance or experienced corporation and they'll help you get started in the right way.

TDLR:

Q: I'm a new EVE-Online player and I've been wondering whether it's viable to earn enough ISK in-game on a regular basis to be able to pay for premium

A: Yes

Q: Is this achievable or will this ruin the fun from the game?

A: Depends on the player, it will likely take away from the fun.

Q: Also is the game fun in the long run without the premium access?

A: Yes, but you'll want it anyway

Q: Do you have any advice for a newbie how to earn ISK?

A: Join a corporation that can teach you how they make ISK.