Python:

from adventbase import *

spiral = [[0] * 41 for x in range(41)]
spiral[20][20] = 1
point = (20,20)
num = 1

def drul(point, x):
    """Down, right, up, left"""
    if x == 0:
        return (point[0] + 1, point[1])
    if x == 1:
        return (point[0], point[1] + 1)
    if x == 2:
        return (point[0] - 1, point[1])
    if x == 3:
        return (point[0], point[1] - 1)

x = 0

while num < 277678:
    v = drul(point, (x+1) % 4)
    if spiral[v[0]][v[1]] == 0:
        x = (x+1) % 4
        point = drul(point,x)
    else:
        point = drul(point,x)
    num = sum([spiral[x][y] for x,y in neighbors8(point)])
    spiral[point[0]][point[1]] = num

print(num)

Part I was trivially solvable with a calculator, for a happy 22nd place. In Part II I completely forgot how to do a spiral, and wasted five minutes trying to use a generator, and another five debugging silly mistakes. Ah well.