There is more information, yes. For example, how long do you need the circuit to run before the battery is used? How quickly do you want to re-charge the battery? What type of colours and patterns and brightness do you want the LEDs to display? All the answers will be needed in order for you to choose a suitable battery.
Although ws2812 is rated for 5V, I have found that they will operate ok at the voltages from Li-ion/Li-po batteries, so you may not require a 5V converter. But there is no guarantee, so you must experiment with your strip & ring.
The current requirements you mentioned above are the maximum that could be required if each led is on full brightness white. Other colours, lower brightness levels, flashing patterns and so on will require considerably less current, giving you much more battery life. For example, an animated rainbow effect will probably require only 33% of that maximum current.
You may be able to charge your battery using the charging circuit on the LoLin board, if it is fast enough for you. It's maximum charging rate is 500mA. This charging circuit will also protect the battery from being overcharged.
You must also protect the battery from being over-discharged. This damages li-ion/Li-po batteries and can make them dangerous, exploding or catching fire when they are re-charged. Some batteries have built-in circuits to prevent over-discharge, but many do not, so you will need to add a circuit/module for that.
With such a high current required to flow from the battery to the strip/ring, you do not want that current to flow through the LoLin board. Use extra wires to take power directly from the battery to the LEDs, bypassing the LoLin board. Separate wires can connect the LoLin board to the battery to power it.