Can't be solvable by DP but by Dijkstra. My solution:

      #include<bits/stdc++.h>
      #define ull unsigned long long
      using namespace std;
      bool is_valid(ull i,ull j,ull n){
              if(i<0 || j<0 || i>=n || j>=n) return false;
              return true;
      }
      int main() {
              ull n;
              cin>>n;
              vector<vector<ull>> a(n, vector<ull>(n, 0));
              for(ull i=0;i<n;i++) for(ull j=0;j<n;j++) cin>>a[i][j];
              vector<pair<ull,ull> > adj[n*n+1];
              for(ull i=0;i<n;i++){
                      for(ull j=0;j<n;j++){
                              if(is_valid(i,j-1,n))adj[i*n+j+1].push_back({a[i][j-1],i*n+j});
                              if(is_valid(i,j+1,n))adj[i*n+j+1].push_back({a[i][j+1],i*n+j+2});
                              if(is_valid(i-1,j,n))adj[i*n+j+1].push_back({a[i-1][j],(i-1)*n+j+1});
                              if(is_valid(i+1,j,n))adj[i*n+j+1].push_back({a[i+1][j],(i+1)*n+j+1});
                      }
              }
              priority_queue<pair<ull,ull>,vector<pair<ull,ull>>,greater<pair<ull,ull>> > pq;
              pq.push({a[0][0],1});
              vector<bool> vis(n*n+1,false);
              vector<ull> dis(n*n+1,LONG_MAX);
              dis[1]=a[0][0];
              while(!pq.empty()){
                      pair<ull,ull> x=pq.top();
                      dis[x.second]=x.first;
                      vis[x.second] = true;
                      pq.pop();
                      for( auto j:adj[x.second]){
                              if(!vis[j.second] && dis[j.second]>x.first+j.first){
                                              dis[j.second]=dis[x.second]+j.first;
                                              pq.push({dis[j.second],j.second});
                                      }
                      }
              }
              cout<<dis[n*n]<<endl;
              return 0;
      }