<select id="part">
<option>
First
</option>
<option>
Second
</option>
<option>
Third
</option>
</select>
<button data-next=1 class='control'>Next</button>
<button class='control'>Previous</button>
const select=document.getElementById('part')
for (let btn of document.querySelectorAll('.control'))
btn.addEventListener( 'click', e => {
let i = select.selectedIndex + ( e.target.dataset.next ? 1 : -1),
l = select.options.length - 1
select.selectedIndex = i > l ? 0 : i < 0 ? l : i
})
Или так, чтобы по кругу не ходил
const select=document.querySelector('#part')
for (let btn of document.querySelectorAll('.control'))
btn.addEventListener( 'click', e => {
let i = select.selectedIndex + ( e.target.dataset.next ? 1 : -1)
if (i in select.options) select.selectedIndex = i
})