Speedy damage Ymir?

It's not a question of being "competitive" it's about the build at least work and make sense, even if you just want to "have fun".

If you want attack speed ymir, pick sands of time, Bancroft, telk, hastaned ring, typhon, poly. Upgrade sands to pendulum and Bancroft to nimble.

You can have great attack speed a lot of sustain with the lifesteal and will melt targets with that much damage

Alright, let's see. I'll comment with that in mind, and also that it's obviously, very obviously a non-meta for-funsies build.

So the basic concept here is obviously based on attack speed more than anything, right? You're looking to get most of your damage from those basics. Where possible, you want to maximize that by choosing items that support basics, with consideration to the need for basic damage being supported by power, attack speed, and penetration. Side consideration is apparently taken to some survivability, based on the presence in your build of Sundering Axe, Berserker's Shield, and Mystical Mail. You've built in a concept of survivability by also choosing the combination of Telkhines Ring and Typhon's Fang for lifesteal function. Does that sound fair and incorporate everything properly?

Then we also consider that you will also be partially reliant on abilities to apply Frostbite, which is absolutely core to getting the most damage you can.

With the above considered, I will tell you that:

• Toxic Blade: should be situational. It's nice that it provides AS and movement speed. In Arena, however, healing abilities and lifesteal really aren't going to be as impactful here as in other modes like Conquest. Base is generally pretty close to quickly go back and heal, so this effect would be for battles of attrition, where you're trying to kill someone before they kill you. In that case, anti-heal can help, but again, should be situational if anything. It's also nice that it can steal AS, so if you're facing multiple basic attackers, it will have more value.
  
  
• Sundering Axe has no synergy with this build. Why are you choosing it? Why wouldn't you go with a basic attacker's starter/upgrade, or at least consider something like Axe of Animosity?
  
  
• Mystical Maill has no synergy with this build. Yes, you can get some use out of it because you can keep enemies fairly close with your wall, your 2's slow, and your 3's freeze, but the damage comparison from this is absolute **** compared to what you might do from good basics. It brings some value from the survivability of phys prots and health, but this item choice just doesn't feel well thought out.
  
  
• Outside of your abilities' ability (ha) to keep enemies close, you'll have a hard time sticking over time. If you're looking to fit in other items that actually synergize, one of the first items that comes to mind is Hastened Ring, for the Haste effect. It also gives you that AS you're looking for.
  
  
• Damage also comes not just from the base damage but also penetration, to help you get through both natural and item-based protections. In your current build, you have 20% from Typhon's and 15 flat from Toxic. Other item considerations to get your pen up might include Demonic Grip. Charon's Coin could be interesting partially for the added MS it can provide, though I probably wouldn't go for it. You might also situationally consider Rod of Tahuti, as it could be interesting once enemies are lower health. But if you're just going minimum, Typhon's is probably semi-okay on its own.
  
  
• An alternative to Telkhines Ring could be grabbing Nimble Bancroft's Talon a bit later. That would situationally give you more power and lifesteal than Telk, but you have to get it later in the build which slightly messes up the flow at the start.
  
  
• If you want magical protection, Shogun's Kusari is an obvious choice for the AS, and any CDR is going to be helpful for a god that's traditionally more ability-based.

Hope these ideas help.